16. Q16
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Similar Question 1
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle y = (\frac{1 + x^2}{1 -x^2})^5</code></p>
Similar Question 2
<img src="/qimages/15309" />
Similar Question 3
<p>Determine the derivative of each function by using the following methods.</p> <ul> <li>i) Use the chain rule, and then simplify.</li> <li>ii) Simplify first, and then differentiate.</li> </ul> <p><code class='latex inline'>p(x)=\sqrt{9x^2}</code></p>
Similar Questions
Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
Now You Try
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p><code class='latex inline'>\displaystyle f(x) = (x - 1)^{-5}</code></p>
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle f(x) = 5\sqrt{4x - 3}</code></p>
<img src="/qimages/15286" /><img src="/qimages/15287" /><p>find <code class='latex inline'>w'(1)</code>.</p>
<p>Find the formulas for derivatives with respect to <code class='latex inline'>x</code> for the function defined in each of the following.</p><p><code class='latex inline'>\displaystyle y = f(g(\sqrt{1 + 3x}))</code></p>
<p>Find the slope of the tangent to the curve at the indicated point in each of the following.</p><p><code class='latex inline'>\displaystyle f(x) = (1 + \sqrt{\frac{x - 1}{2}})^{4}</code> at (1, 1)</p>
<p>Differentiate, expressing each answer using positive exponents.</p><p><code class='latex inline'>y=(4x+1)^2</code></p>
<img src="/qimages/15263" />
<img src="/qimages/15272" />
<p>Determine an equation for the tangent to the curve <code class='latex inline'>y=(x^3-4x^2)^3</code> at <code class='latex inline'>x=3</code>.</p>
<img src="/qimages/15308" />
<img src="/qimages/15239" />
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle y = \frac{\sqrt{x + 1}}{x + 2}</code></p>
<p>Find the function <code class='latex inline'>F(x)</code> if the point <code class='latex inline'>(1, 3)</code> is on the graph and </p><p><code class='latex inline'> \displaystyle F'(x) = (2x^2 + \frac{1}{x^4})(x - \sqrt{x}) </code></p>
<img src="/qimages/15278" /><p>Find an equation of the tangent line to the curve at the given point.</p>
<img src="/qimages/15269" />
<img src="/qimages/15280" /><p>Do part b) only.</p>
<img src="/qimages/15252" />
<p>Determine <code class='latex inline'>\displaystyle{\frac{d^2y}{dx^2}}</code> for the function <code class='latex inline'>y=\sqrt{2x+1}</code>.</p>
<img src="/qimages/15297" />
<p>(a) Use two different methods to differentiate <code class='latex inline'>f(x)=\sqrt{25x^4}</code>.</p><p>(b) Use two different methods to differentiate <code class='latex inline'>f(x)=\sqrt{25x^4}</code>.</p>
<img src="/qimages/15299" />
<p>Determine <code class='latex inline'>f'(1)</code>.</p><p><code class='latex inline'>f(x)=\sqrt{4x^2+1}</code></p>
<p>The owners of Mooses, Gooses, and Juices are interested in analysing the productivity of their staff. The function <code class='latex inline'>N(t)=150-\displaystyle{\frac{600}{\sqrt{16+3t^2}}}</code> models the total number, <code class='latex inline'>N</code>, of customers served by the staff after <code class='latex inline'>t</code> hours during an <code class='latex inline'>8h</code> workday <code class='latex inline'>(0\leq t\leq8)</code></p><p>(c) Solve <code class='latex inline'>N(t)=103</code>. Interpret your answer for this situation.</p>
<img src="/qimages/15237" />
<p>Determine the derivative of each function by using the following methods.</p> <ul> <li>i) Use the chain rule, and then simplify.</li> <li>ii) Simplify first, and then differentiate.</li> </ul> <p><code class='latex inline'>p(x)=\sqrt{9x^2}</code></p>
<img src="/qimages/15290" />
<img src="/qimages/15259" />
<p>Express each function as a power with a rational exponent and then differentiate. Express each answer using positive exponents.</p><p><code class='latex inline'>y=\sqrt[3]{x-x^4}</code></p>
<p>If <code class='latex inline'>\displaystyle f(x) = g[h(x)] = (2 -x^3)^4</code>, find $g(x), h(x), h&#39;(x), g&#39;[h(x)], f&#39;(x)$`</p>
<img src="/qimages/15294" /><img src="/qimages/15295" />
<img src="/qimages/15312" />
<img src="/qimages/15288" /><img src="/qimages/15289" /><p>Find h&#39;(2)</p>
<p>The red squirrel population, <code class='latex inline'>p</code>, in a neighbourhood park can be modelled by the function <code class='latex inline'>p(t)=\sqrt{210t+44t^2}</code>, where <code class='latex inline'>t</code> is time, in years.</p> <ul> <li>When is the instantaneous rate of change of the squirrel population approximately 7 squirrels per year?</li> </ul>
<p>Find the derivative of each of the following. </p><p><code class='latex inline'>\displaystyle f(x) = \frac{1}{\sqrt{x^2 - 7x + 12}}</code></p>
<img src="/qimages/15302" />
<img src="/qimages/15282" />
<img src="/qimages/15251" />
<p>Find <code class='latex inline'>a</code> so that the curve <code class='latex inline'>y = \sqrt{ax^2 -4}</code> has a tangent with slope 2 at the point where <code class='latex inline'>x = 2</code>.</p>
<p>Determine the point(s) on the curve <code class='latex inline'>y=x^2(x^3-x)^2</code> where the tangent line is horizontal.</p>
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle f(x) = \sqrt{4- x^2}</code></p>
<p>Differentiate, expressing each answer using positive exponents.</p><p><code class='latex inline'>y=(x^3-x)^{-3}</code></p>
<p>Find the formulas for derivatives with respect to <code class='latex inline'>x</code> for the function defined in each of the following.</p><p><code class='latex inline'>\displaystyle y = f(g(h(x)))</code></p>
<img src="/qimages/15248" />
<img src="/qimages/15262" />
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle f(x) = -2\sqrt{x^2 + 4x}</code></p>
<p>Find the formulas for derivatives with respect to <code class='latex inline'>x</code> for the function defined in each of the following.</p><p><code class='latex inline'>\displaystyle y = f^3(g(x+ 1))</code></p>
<p>Find the derivative of each of the following. </p><p><code class='latex inline'>\displaystyle y = [(x^4 + 1)^3 +1]^5</code></p>
<p>Differentiate. Do NOT simplify.</p><p><code class='latex inline'> \displaystyle y = \sqrt{4x^2 - 9}(2x - 5)^4 </code></p>
<img src="/qimages/15266" />
<p>The owners of Mooses, Gooses, and Juices are interested in analysing the productivity of their staff. The function <code class='latex inline'>N(t)=150-\displaystyle{\frac{600}{\sqrt{16+3t^2}}}</code> models the total number, <code class='latex inline'>N</code>, of customers served by the staff after <code class='latex inline'>t</code> hours during an <code class='latex inline'>8h</code> workday <code class='latex inline'>(0\leq t\leq8)</code></p><p>(d) Determine <code class='latex inline'>N'(t)</code> for the value you found in part c). Compare this value with <code class='latex inline'>N'(4)</code>. What conclusion, if any, can be made from comparing these two values?</p>
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle f(x) = (x^2 -x - 1)^{-2}</code></p>
<p>If <code class='latex inline'>\displaystyle f(x) = g[h(x)] = \sqrt{x^4-3x^2}</code>, find $g(x), h(x), h&#39;(x), g&#39;[h(x)], f&#39;(x)$`</p>
<img src="/qimages/15285" />
<img src="/qimages/15274" />
<img src="/qimages/15256" />
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle f(x) = (x^2-x-1)^{-2}</code></p>
<img src="/qimages/15281" />
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle f(x) = \sqrt{x - 1}</code></p>
<p>Find the slope of the tangent line to the curve <code class='latex inline'>y =f(x)</code>, at the given point P. </p><p><code class='latex inline'>\displaystyle f(x) = \sqrt{\frac{1- x}{1 + x}}</code> at P(0, 1).</p>
<p>If <code class='latex inline'>f(x)=x^2</code>, <code class='latex inline'>g(x)=\displaystyle{\frac{1}{x}}</code>, and <code class='latex inline'>h(x)=\sqrt{x^2+2x}</code>, determine the derivative of each composite function</p><p><code class='latex inline'>y=f\circ g\circ h (x)</code></p>
<img src="/qimages/15286" /><img src="/qimages/15287" /><p>find <code class='latex inline'>v'(1)</code>.</p>
<img src="/qimages/15270" />
<img src="/qimages/15240" />
<p>The population, <code class='latex inline'>P</code>, of a small town can be modelled by the function <code class='latex inline'>P(t)=\displaystyle{\frac{1250}{1+0.01t}}</code>, where <code class='latex inline'>t</code> is time, in years, <code class='latex inline'>t\geq0</code>. Determine the instantaneous rate of change of the population at 2 years, 4 years, and 7 years.</p>
<p>Express each of the following as a power with a negative exponent, and then differentiate. Express each answer using positive exponents.</p><p><code class='latex inline'>y=\displaystyle{\frac{1}{\sqrt{x^2+4x}}}</code></p>
<p>(c) Can both methods that you described in part a) be used to differentiate <code class='latex inline'>f(x)=\sqrt{25x^4-3}</code>? Explain.</p>
<img src="/qimages/15305" /><img src="/qimages/15307" />
<p>Using Leibniz notation, apply the chain rule to determine <code class='latex inline'>\displaystyle{\frac{dy}{dx}}</code> at the indicated value of <code class='latex inline'>x</code>.</p><p><code class='latex inline'>y=\sqrt{u}</code>, <code class='latex inline'>u=2x^2+3x+4</code>, <code class='latex inline'>x=-3</code></p>
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle y = (\frac{1 + x^2}{1 -x^2})^5</code></p>
<p>Determine the derivative of each function by using the following methods.</p> <ul> <li>i) Use the chain rule, and then simplify.</li> <li>ii) Simplify first, and then differentiate.</li> </ul> <p><code class='latex inline'>f(x)=(2x)^3</code></p>
<img src="/qimages/15284" />
<img src="/qimages/15246" />
<p>The red squirrel population, <code class='latex inline'>p</code>, in a neighbourhood park can be modelled by the function <code class='latex inline'>p(t)=\sqrt{210t+44t^2}</code>, where <code class='latex inline'>t</code> is time, in years.</p><p>(b) When will the population reach 60 squirrels?</p>
<img src="/qimages/15265" />
<p>Determine an equation for the tangent to the curve <code class='latex inline'>\displaystyle{\frac{1}{\sqrt[5]{5x^3-2x^2}}}</code> at <code class='latex inline'>x=2</code>.</p>
<img src="/qimages/15279" /><p>Do part b) only</p>
<p>The position function of a moving particle is <code class='latex inline'>s(t)=\sqrt[3]{t^5-750t^2}</code>, where <code class='latex inline'>s</code> is in metres and <code class='latex inline'>t</code> is in seconds. Determine the velocity of the particle at <code class='latex inline'>5s</code>.</p>
<p>Determine <code class='latex inline'>f'(1)</code>.</p><p><code class='latex inline'>f(x)=(3-x+x^2)^{-2}</code></p>
<img src="/qimages/15303" />
<p>Express <code class='latex inline'>y=\displaystyle{\frac{4x-x^3}{(3x^2+2)^2}}</code> as a product and then differentiate. Simplify your answer using positive exponents.</p>
<img src="/qimages/15311" />
<p>Find the slope of the tangent to the curve at the indicated point in each of the following.</p><p><code class='latex inline'>\displaystyle f(x) = (1 - \frac{4}{x + 2})^3</code> at (0, -1)</p>
<img src="/qimages/15264" />
<p>Find the formulas for derivatives with respect to <code class='latex inline'>x</code> for the function defined in each of the following.</p><p><code class='latex inline'>\displaystyle y= f(g^2(x))</code></p>
<img src="/qimages/15236" />
<img src="/qimages/15250" />
<p>Express each function as a power with a rational exponent and then differentiate. Express each answer using positive exponents.</p><p><code class='latex inline'>y=\sqrt[5]{2+3x^2-x^3}</code></p>
<img src="/qimages/15298" />
<img src="/qimages/15305" /><img src="/qimages/15306" />
<img src="/qimages/15277" /><p>Find an equation of the tangent line to the curve at the given point.</p>
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle y = (2x^2 -3x + 1)^{\frac{1}{2}}</code></p>
<p>The red squirrel population, <code class='latex inline'>p</code>, in a neighbourhood park can be modelled by the function <code class='latex inline'>p(t)=\sqrt{210t+44t^2}</code>, where <code class='latex inline'>t</code> is time, in years.</p> <ul> <li>What is the instantaneous rate of change of the population at the time in part b)?</li> </ul>
<img src="/qimages/15260" />
<img src="/qimages/15291" />
<img src="/qimages/15301" />
<img src="/qimages/15293" />
<img src="/qimages/15286" /><img src="/qimages/15287" /><p>find <code class='latex inline'>u'(1)</code>.</p>
<img src="/qimages/15276" /><p>Find an equation of the tangent line to the curve at the given point.</p>
<img src="/qimages/15245" />
<p>Find the derivative.</p><p><code class='latex inline'>\displaystyle y =\ sqrt{1 + xe^{-2x}} </code></p>
<p>This figure shows a graph of <code class='latex inline'>y=f(x)</code>.</p><img src="/qimages/1074" /><p>If the function <code class='latex inline'>F</code> is defined by <code class='latex inline'>F(x)=f[f(x)]</code>, then <code class='latex inline'>F(1)</code> equals</p> <ul> <li><strong>A</strong> -1</li> <li><strong>B</strong> 2</li> <li><strong>C</strong> 4</li> <li><strong>D</strong> 4.5</li> <li><strong>E</strong> undefined</li> </ul>
<p>If <code class='latex inline'>\displaystyle f(x) = g[h(x)] = (7 + x^2)^{-2}</code>, find $g(x), h(x), h&#39;(x), g&#39;[h(x)], f&#39;(x)$`</p>
<img src="/qimages/15255" />
<p>Differentiate. Do NOT simplify.</p><p><code class='latex inline'> \displaystyle y = \big((4x + 1)^5+\sqrt{1 + 2x + 3x^2}\big)^{-\frac{2}{3}} </code></p>
<p>If <code class='latex inline'>\displaystyle f(x) = g[h(x)] = (-3x + 4)^{-1}</code>, find $g(x), h(x), h&#39;(x), g&#39;[h(x)], f&#39;(x)$`</p>
<img src="/qimages/15267" />
<p>Find the derivative of each of the following. </p><p><code class='latex inline'>\displaystyle f(x) = \sqrt{1 + (x^2 + 1)^3}</code> </p>
<p>Find an equation of the tangent line to the curve at the given point.</p><img src="/qimages/15275" />
<p>Using Leibniz notation, apply the chain rule to determine <code class='latex inline'>\displaystyle{\frac{dy}{dx}}</code> at the indicated value of <code class='latex inline'>x</code>.</p><p><code class='latex inline'>y=u(2-u^2)</code>, <code class='latex inline'>u=\displaystyle{\frac{1}{x}}</code>, <code class='latex inline'>x=2</code></p>
<p>Find the derivative.</p><p><code class='latex inline'>\displaystyle f(x) = \sin^2(e^{\sin^2x}) </code></p>
<p>Determine the derivative of each function by using the following methods.</p> <ul> <li>i) Use the chain rule, and then simplify.</li> <li>ii) Simplify first, and then differentiate.</li> </ul> <p><code class='latex inline'>g(x)=(-4x^2)^2</code></p>
<img src="/qimages/15258" />
<img src="/qimages/15279" /><p>Do part a) only</p>
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle f(x) = (3x - 5)^{\frac{2}{3}}</code></p>
<p>Express each of the following as a power with a negative exponent, and then differentiate. Express each answer using positive exponents.</p><p><code class='latex inline'>y=\displaystyle{\frac{1}{(3x^2-2)}}</code></p>
<img src="/qimages/15292" />
<p>Find the derivative.</p><p><code class='latex inline'>\displaystyle y = \sqrt{x + \sqrt{x + \sqrt{x}}} </code></p>
<img src="/qimages/15247" />
<p>Determine <code class='latex inline'>f'(1)</code>.</p><p><code class='latex inline'>f(x)=(4x^2-x+1)^2</code></p>
<img src="/qimages/15309" />
<img src="/qimages/15310" />
<p>Differentiate. Do NOT simplify.</p><p><code class='latex inline'> \displaystyle y = \frac{\sqrt[3]{4x + 3}}{(5x^2 - 3x)^3} </code></p>
<img src="/qimages/15243" />
<img src="/qimages/15261" />
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle f(x) = (3x^3 -1)^{-\frac{5}{2}}</code></p>
<img src="/qimages/15300" />
<img src="/qimages/15288" /><img src="/qimages/15289" /><p>Find g&#39;(2)</p>
<img src="/qimages/15244" />
<p>Find the formulas for derivatives with respect to <code class='latex inline'>x</code> for the function defined in each of the following.</p><p><code class='latex inline'>\displaystyle y = (f(g(x)))^4</code></p>
<img src="/qimages/15253" />
<img src="/qimages/15238" />
<p>Differentiate, expressing each answer using positive exponents.</p><p><code class='latex inline'>y=(3x^2-2)^3</code></p>
<img src="/qimages/15235" />
<p>Find the slope of the tangent to the curve at the indicated point in each of the following.</p><p> <code class='latex inline'>\displaystyle f(x) = \sqrt{9 + x^2}</code> at (4, 5)</p>
<p>Find the derivative of each of the following. </p><p><code class='latex inline'>\displaystyle f(x) =(1 - \sqrt{x^3 + 1})^5 </code></p>
<img src="/qimages/15254" />
<p>Express each function as a power with a rational exponent and then differentiate. Express each answer using positive exponents.</p><p><code class='latex inline'>y=\sqrt{2x-3x^5}</code></p>
<img src="/qimages/15257" />
<img src="/qimages/15283" />
<img src="/qimages/15304" />
<p>Determine the derivative of each function by using the following methods.</p> <ul> <li>i) Use the chain rule, and then simplify.</li> <li>ii) Simplify first, and then differentiate.</li> </ul> <p><code class='latex inline'>f(x)=(16x^2)^{\frac{3}{4}}</code></p>
<p>The owners of Mooses, Gooses, and Juices are interested in analysing the productivity of their staff. The function <code class='latex inline'>N(t)=150-\displaystyle{\frac{600}{\sqrt{16+3t^2}}}</code> models the total number, <code class='latex inline'>N</code>, of customers served by the staff after <code class='latex inline'>t</code> hours during an <code class='latex inline'>8h</code> workday <code class='latex inline'>(0\leq t\leq8)</code></p><p>(b) Determine <code class='latex inline'>N(4)</code> and <code class='latex inline'>N'(4)</code>. Interpret each of these values for this situation.</p>
<p>Find the slope of the tangent to the curve at the indicated point in each of the following.</p><p><code class='latex inline'>\displaystyle y = (1- x^3)^2</code> at (1, 0)</p>
<p>Express each function as a power with a rational exponent and then differentiate. Express each answer using positive exponents.</p><p><code class='latex inline'>y=\sqrt{-x^3+9}</code></p>
<p>Using Leibniz notation, apply the chain rule to determine <code class='latex inline'>\displaystyle{\frac{dy}{dx}}</code> at the indicated value of <code class='latex inline'>x</code>.</p><p><code class='latex inline'>y=u^2+3u</code>, <code class='latex inline'>u=\sqrt{x}</code>, <code class='latex inline'>x=4</code></p>
<img src="/qimages/15271" />
<p>Let <code class='latex inline'>f</code> be a function such that <code class='latex inline'>f'(x)=\displaystyle{\frac{1}{x}}</code>. What is the derivative <code class='latex inline'>(f^{-1})'(x)</code> of its inverse? Hint: If <code class='latex inline'>y=f^{-1}(x)</code>, you can write <code class='latex inline'>f(y)=x</code>.</p> <ul> <li><strong>A</strong> 0</li> <li><strong>B</strong> <code class='latex inline'>x</code></li> <li><strong>C</strong> <code class='latex inline'>-\displaystyle{\frac{1}{x^2}}</code></li> <li><strong>D</strong> <code class='latex inline'>f(x)</code></li> <li><strong>E</strong> <code class='latex inline'>f^{-1}(x)</code></li> </ul>
<img src="/qimages/15296" />
<p>The red squirrel population, <code class='latex inline'>p</code>, in a neighbourhood park can be modelled by the function <code class='latex inline'>p(t)=\sqrt{210t+44t^2}</code>, where <code class='latex inline'>t</code> is time, in years.</p><p>(a) Determine the rate of growth of the squirrel population at <code class='latex inline'>t = 2</code> years.</p>
<p>If <code class='latex inline'>\displaystyle f(x) = g[h(x)] = (6x -1)^2</code>, find <code class='latex inline'>g(x), h(x), h'(x), g'[h(x)], f'(x)</code></p>
<p>Using Leibniz notation, apply the chain rule to determine <code class='latex inline'>\displaystyle{\frac{dy}{dx}}</code> at the indicated value of <code class='latex inline'>x</code>.</p><p><code class='latex inline'>y=\displaystyle{\frac{1}{u^2}}</code>, <code class='latex inline'>u=x^3-5x</code>, <code class='latex inline'>x=-2</code></p>
<p>Express each of the following as a power with a negative exponent, and then differentiate. Express each answer using positive exponents.</p><p><code class='latex inline'>y=\displaystyle{\frac{1}{(-x^3+1)^2}}</code></p>
<p>Determine equations for the tangents to the curve <code class='latex inline'>y=x^3\sqrt{8x^2+1}</code> at the points where <code class='latex inline'>x=1</code> and <code class='latex inline'>x=-1</code>. How are the tangent lines related? Explain why this relationship is true at all points with corresponding positive and negative values <code class='latex inline'>x = a</code> and <code class='latex inline'>x = -a</code></p>
<p>Find the slope of the tangent to the curve at the indicated point in each of the following.</p><p><code class='latex inline'>\displaystyle f(x) = \sqrt{1 +x^3}</code> at (2, 3)</p>
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle f(x) = (x^2 -3)^{11}</code></p>
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle y = (x + \frac{1}{x})^5</code></p>
<p>If <code class='latex inline'>\displaystyle f(x) = g[h(x)] = (x^2 + 3)^3</code>, find $g(x), h(x), h&#39;(x), g&#39;[h(x)], f&#39;(x)$`</p>
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle y= \frac{(3x^2 -1)^{-4}}{(x^3 -2x)^{-5}}</code></p>
<p>Differentiate, expressing each answer using positive exponents.</p><p><code class='latex inline'>y=(4x^2+3x)^{-2}</code></p>
<img src="/qimages/15249" />
<p>The owners of Mooses, Gooses, and Juices are interested in analysing the productivity of their staff. The function <code class='latex inline'>N(t)=150-\displaystyle{\frac{600}{\sqrt{16+3t^2}}}</code> models the total number, <code class='latex inline'>N</code>, of customers served by the staff after <code class='latex inline'>t</code> hours during an <code class='latex inline'>8h</code> workday <code class='latex inline'>(0\leq t\leq8)</code></p><p>(a) Determine <code class='latex inline'>N'(t)</code>. What does the derivative represent for this situation?</p>
<img src="/qimages/15313" />
<img src="/qimages/15273" />
<p>Determine <code class='latex inline'>f'(1)</code>.</p><p><code class='latex inline'>f(x)=\displaystyle{\frac{5}{\sqrt[3]{2x-x^2}}}</code></p>
<p>Find y&#39; and y&#39;&#39;.</p><p><code class='latex inline'>\displaystyle y = \cos(\sin 3\theta) </code></p>
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle f(x) = (2x^2 -3x + 1)^{\frac{1}{2}}</code></p>
<p>Use the chain rule to find the derivative of the functions defined in each of the following.</p><p> <code class='latex inline'>\displaystyle y = (x^2 + 4)\sqrt{x^2 + 2}</code></p>
<img src="/qimages/15241" />
<p>Find the formulas for derivatives with respect to <code class='latex inline'>x</code> for the function defined in each of the following.</p><p> <code class='latex inline'>\displaystyle y = f(\sqrt{1 + g(x -3)})</code></p>
<p>Determine the derivative of each function by using the following methods.</p> <ul> <li>i) Use the chain rule, and then simplify.</li> <li>ii) Simplify first, and then differentiate.</li> </ul> <p><code class='latex inline'>q(x)=(8x)^{\frac{2}{3}}</code></p>
<img src="/qimages/15280" /><p>Do part a) only.</p>
<p>Express each of the following as a power with a negative exponent, and then differentiate. Express each answer using positive exponents.</p><p><code class='latex inline'>y=\displaystyle{\frac{1}{\sqrt[3]{x-7x^2}}}</code></p>
<img src="/qimages/15242" />
<p>The formula for the volume of a cube in terms of its side length, <code class='latex inline'>s</code>, is <code class='latex inline'>V(s)=s^3</code>. If the side length is expressed in terms of a variable, <code class='latex inline'>x</code>,such that <code class='latex inline'>s=3x^2-7x+1</code>, determine <code class='latex inline'>\displaystyle{\frac{dV}{dx}}\displaystyle{|_{x=3}}</code>. Interpret this value for this situation.</p>
<p>Find the slope of the tangent to the curve at the indicated point in each of the following.</p><p><code class='latex inline'>\displaystyle f(x) = (\frac{x}{x + 1})^2</code> at (0, 0)</p>
<img src="/qimages/15268" />
<p>Find the derivative of each of the following. </p><p><code class='latex inline'>\displaystyle f(x) = \sqrt{1 + \sqrt{x}}</code></p>
<p>Find the derivative of each of the following. </p><p><code class='latex inline'>\displaystyle f(x) = [(x^4 + 1)^3 + 1]^{\frac{1}{2}}</code></p>
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