Chapter Review Derivative Rules
Chapter
Chapter 2
Section
Chapter Review Derivative Rules
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Solutions 67 Videos

Use the definition of the derivative to find f'(x) for each of the following functions.

y=2x^2-5x

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1.38mins
Q2a

Differentiate each of the following functions:

y=x^2-5x+4

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0.14mins
Q3a

Differentiate each of the following functions:

\displaystyle{f(x)=x^{\displaystyle{\frac{3}{4}}}}

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0.27mins
Q3b

Differentiate each of the following functions:

\displaystyle{y=\frac{7}{3x^4}}

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0.28mins
Q3c

Differentiate each of the following functions:

\displaystyle{y=\frac{1}{x^2+5}}

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0.23mins
Q3d

Differentiate each of the following functions:

\displaystyle{y=\frac{3}{(3-x^2)^2}}

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0.51mins
Q3e

Differentiate each of the following functions:

\sqrt{7x^2+4x+1}

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0.47mins
Q3f

Determine the derivative of the given function. In some cases, it will save time if you rearrange the function before differentiating.

f(x)=\displaystyle{\frac{2x^3-1}{x^2}}

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0.30mins
Q4a

Determine the derivative of the given function. In some cases, it will save time if you rearrange the function before differentiating.

g(x)=\sqrt{x}(x^3-x)

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0.18mins
Q4b

Determine the derivative of the given function. In some cases, it will save time if you rearrange the function before differentiating.

\displaystyle{y=\frac{x}{3x-5}}

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0.25mins
Q4c

Determine the derivative of the given function. In some cases, it will save time if you rearrange the function before differentiating.

y=\sqrt{x-1}(x+1)

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0.41mins
Q4d

Determine the derivative of the given function. In some cases, it will save time if you rearrange the function before differentiating.

f(x)=(\sqrt{x}+2)^{\displaystyle{\frac{-2}{3}}}

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0.39mins
Q4e

Determine the derivative of the given function. In some cases, it will save time if you rearrange the function before differentiating.

\displaystyle{y=\frac{x^2+5x+4}{x+4}}

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0.24mins
Q4f

Determine the derivative, and give your answer in a simplified form.

y=x^4(2x-5)^6

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1.18mins
Q5a

Determine the derivative, and give your answer in a simplified form.

y=x\sqrt{x^2+1}

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0.52mins
Q5b

Determine the derivative, and give your answer in a simplified form.

y=\displaystyle{\frac{(2x-5)^4}{(x+1)^3}}

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1.42mins
Q5c

Determine the derivative, and give your answer in a simplified form.

y=\displaystyle{\left(\frac{10x-1}{3x+5} \right)^6}

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1.21mins
Q5d

Determine the derivative, and give your answer in a simplified form.

y=(x-2)^3(x^2+9)^4

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1.21mins
Q5e

Determine the derivative, and give your answer in a simplified form.

y=(1-x^2)^3(6+2x)^{-3}

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2.06mins
Q5f

If f is a differentiable function, find an expression for the derivative of each of the following functions:

g(x)=f(x^2)

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0.10mins
Q6a

If f is a differentiable function, find an expression for the derivative of each of the following functions:

h(x)=2xf(x)

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0.27mins
Q6b

If y=5u^2+3u-1 and u=\displaystyle{\frac{18}{x^2+5}}, find \displaystyle{\frac{dy}{dx}} when x=2.

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1.32mins
Q7a

If y=\displaystyle{\frac{u+4}{u-4}} and u=\displaystyle{\frac{\sqrt{x}+x}{10}}, find \displaystyle{\frac{dy}{dx}} when x=4.

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1.57mins
Q7b

If y=f(\sqrt{x^2+9}) and f'(5)=-2, find \displaystyle{\frac{dy}{dx}} when x=4.

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1.12mins
Q7c

Determine the slope of the tangent at point (1,4) on the graph of f(x)=(9-x^2)^{\displaystyle{\frac{2}{3}}}.

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0.42mins
Q8

a. For what values of x does the curve y=-x^3+6x^2 have a slope of -12?

b. For what values of x does the curve y=-x^3+6x^2 have a slope of -15?

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1.32mins
Q9

Determine the values of x where the graph of each function has a horizontal tangent.

i. y=(x^2-4)^5

ii. \displaystyle y = (x^3 - x)^2

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1.18mins
Q10a

Determine the equation of the tangent to each function at the given point.

y=(x^2+5x+2)^4, (0,16)

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0.43mins
Q11a

Determine the equation of the tangent to each function at the given point.

y=(3x^{-2}-2x^3)^5, (1,1)

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0.46mins
Q11b

A tangent to the parabola y=3x^2-7x+5 is perpendicular to x+5y-10=0. Determine the equation of the tangent.

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1.17mins
Q12

The line y=8x+b is tangent to the curve y=2x^2. Determine the point of tangency and the value of b.

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0.45mins
Q13

Consider the function f(x)=2x^{\displaystyle{\frac{5}{3}}}-5x^{\displaystyle{\frac{2}{3}}}

Determine the slope of the tangent at the point where the graph crosses the x-axis.

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1.44mins
Q15a

Consider the function f(x)=2x^{\displaystyle{\frac{5}{3}}}-5x^{\displaystyle{\frac{2}{3}}}

  • Determine the value of a shown in the graph of f(x) given below.
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0.21mins
Q15b

A rested student is able to memorize M words after t minutes, where M=0.1t^2-0.001t^3, 0\leq t\leq \displaystyle{\frac{200}{3}}.

i. How many words are memorized in the first 10 min?

ii. How many words are memorized in the first 15 min?

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0.53mins
Q16a

A rested student is able to memorise M words after t minutes, where M=0.1t^2-0.001t^3, 0\leq t\leq \displaystyle{\frac{200}{3}}.

i. What is the memory rate at t=10?

ii. What is the memory rate at t=15?

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1.08mins
Q16b

A grocery store determines that, after t hours on the job, a new cashier can scan N(t)=20-\displaystyle{\frac{30}{\sqrt{9+t^2}}} items per minute.

a) Find N'(t), the rate at which the cashier's productivity is changing.

b) According to this model, does the cashier ever stop improving? Why?

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1.48mins
Q17

An athletic-equipment supplier experiences weekly costs of C(x)=\displaystyle{\frac{1}{3}}x^3+40x+700in producing x baseball gloves per week.

Find the marginal cost, C'(x)

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0.19mins
Q18

An athletic-equipment supplier experiences weekly costs of C(x)=\displaystyle{\frac{1}{3}}x^3+40x+700in producing x baseball gloves per week.

  • Find the production level x at which the marginal cost is $76 per glove.
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0.24mins
Q18b

A manufacturer of kitchen appliances experiences revenue of R(x)750x-\displaystyle{\frac{x^2}{6}}-\displaystyle{\frac{2}{3}}x^3 dollars from the sale of x refrigerators per month.

  • Find the marginal revenue, R'(x).
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0.25mins
Q19a

A manufacturer of kitchen appliances experiences revenue of R(x)750x-\displaystyle{\frac{x^2}{6}}-\displaystyle{\frac{2}{3}}x^3 dollars from the sale of x refrigerators per month.

  • Find the marginal revenue when 10 refrigerators per month are sold.
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0.21mins
Q19b

An economist has found that the demand function for a particular new product is given by D(p)=\displaystyle{\frac{20}{\sqrt{p-1}}}, p>1. Find the slope of the demand curve at the point (5,10).

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1.14mins
Q20

Kathy has diabetes. Her blood sugar level, B, one hour after an insulin injection, depends on the amount of insulin, x in milligrams injected. B(x)=-0.2x^2+500, 0\leq x\leq 40.

  • Find B(0) and B(30).
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0.27mins
Q21a

Kathy has diabetes. Her blood sugar level, B, one hour after an insulin injection, depends on the amount of insulin, x in milligrams injected. B(x)=-0.2x^2+500, 0\leq x\leq 40.

  • Find B'(0) and B'(30)
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0.28mins
Q21b

Kathy has diabetes. Her blood sugar level, B, one hour after an insulin injection, depends on the amount of insulin, x in milligrams injected. B(x)=-0.2x^2+500, 0\leq x\leq 40.

  • Find B'(0) and B'(30) and interpret your results.
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0.21mins
Q21c

Kathy has diabetes. Her blood sugar level, B, one hour after an insulin injection, depends on the amount of insulin, x in milligrams injected. B(x)=-0.2x^2+500, 0\leq x\leq 40.

  • Find the values of B'(50) and B(50).
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1.01mins
Q21d

Determine if the function is differentiable at x=1.

m(x)=|3x-3|-1

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1.19mins
Q22d

At what x-values is each function not differentiable? Explain.

f(x)=\displaystyle{\frac{3}{4x^2-x}}

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0.17mins
Q23a

At what x-values is each function not differentiable? Explain.

f(x)=\displaystyle{\frac{x^2-x-6}{x^2-9}}

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0.31mins
Q23b

At what x-values is each function not differentiable? Explain.

f(x)=\sqrt{x^2-7x+6}

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0.50mins
Q23c

At a manufacturing plant, productivity is measured by the number of items, p, produced per employee per day over the previous 10 years. Productivity is modelled by p(t)=\displaystyle{\frac{25t}{t+1}}, where t is the number of years measured from 10 years ago. Determine the rate of change of p with respect to t.

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0.32mins
Q24

Given f(x)=\frac{(2x-3)^2+5}{2x-3},

Express f as the composition of two simpler functions.

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0.32mins
Q26a

Given f(x)=\frac{(2x-3)^2+5}{2x-3} and y = u+5u^{-1}, u = 2x -3

Use this composition to determine f'(x)

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0.50mins
Q26b

Given g(x)=\sqrt{2x-3}+5(2x-3),

  • Express g as the composition of two simpler functions.
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0.39mins
Q27a

Given g(x)=\sqrt{2x-3}+5(2x-3),

Use this composition to determine g'(x).

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0.28mins
Q27b

Determine the derivative of the function.

f(x)=(2x-5)^3(3x^2+4)^5

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1.46mins
Q28a

Determine the derivative of the function.

g(x)=(8x^3)(4x^2+2x-3)^5

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1.52mins
Q28b

Determine the derivative of the function.

y=(5+x)(4-7x^3)^6

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2.25mins
Q28c

Determine the derivative of the function.

h(x) = \frac{6x - 1}{(3x + 5)^4}

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1.27mins
Q28d

Determine the derivative of the function.

y=\displaystyle{\frac{(2x^2-5)}{(x+8)^2}}

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1.29mins
Q28e

Determine the derivative of the function.

f(x)=\displaystyle{\frac{-3x^4}{\sqrt{4x-8}}}

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2.09mins
Q28f

Determine the derivative of the function.

g(x)=\displaystyle{\left( \frac{2x+5}{6-x^2} \right)^4}

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1.51mins
Q28g

Determine the derivative of the function.

y=\displaystyle{\left[ \frac{1}{(4x+x^2)^3}\right]^3}

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0.45mins
Q28h

Find numbers a, b, and c so that the graph f(x)=ax^2+bx+c has x-intercepts at (0,0) and (8,0), and a tangent with slope 16 where x=2.

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1.24mins
Q29

An ant colony was treated with an insecticide and the number of survivors, A, in hundreds at t hours is A(t)=-t^3+5t+750.

Find A'(t).

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0.07mins
Q30a

An ant colony was treated with an insecticide and the number of survivors, A, in hundreds at t hours is A(t)=-t^3+5t+750.

Find the rate of change of the number of living ants in the colony at 5 h.

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0.17mins
Q30b

An ant colony was treated with an insecticide and the number of survivors, A, in hundreds at t hours is A(t)=-t^3+5t+750.

  • How many ants were in the colony before it was treated with the insecticide?
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0.10mins
Q30c

An ant colony was treated with an insecticide and the number of survivors, A, in hundreds at t hours is A(t)=-t^3+5t+750.

How many hours after the insecticide was applied were no ants remaining in the colony?

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0.31mins
Q30d