Differentiate each of the following:
y=6-e^x
Differentiate each of the following:
y=2x+3e^x
Differentiate each of the following:
y=e^{2x+3}
Differentiate each of the following:
y=e^{-3x^2+5x}
Differentiate each of the following:
y=xe^x
Differentiate each of the following:
s=\displaystyle{\frac{e^t-1}{e^t+1}}
Determine $\displaystyle{\frac{dy}{dx}}$ for each of the following:
y=10^x
Determine \displaystyle{\frac{dy}{dx}} for each of the following:
y=4^{3x^2}
Determine \displaystyle{\frac{dy}{dx}} for each of the following:
y=(5x)(5^x)
Determine \displaystyle{\frac{dy}{dx}} for each of the following:
y=(x^4)2^x
Determine \displaystyle{\frac{dy}{dx}} for each of the following:
y=\displaystyle{\frac{4x}{4^x}}
Determine \displaystyle{\frac{dy}{dx}} for each of the following:
y=\displaystyle{\frac{5^{\sqrt{x}}}{x}}
Differentiate each of the following:
y=3\sin2x-4\cos2x
Differentiate each of the following:
y=\tan3x
Differentiate each of the following:
y=\displaystyle{\frac{1}{2-\cos x}}
Differentiate each of the following:
y=x\tan2x
Differentiate each of the following:
y=(\sin2x)e^{3x}
Differentiate each of the following:
y=\cos^22x
a) Given the function f(x)=\displaystyle{\frac{e^x}{x}}, solve the equation f'(x)=0.
b) Discuss the significance of the solution you found in part a.
If f(x)=xe^{-2x}, find f'\left(\displaystyle{\frac{1}{2}}\right).
a) If f(x)=xe^{-2x}, find f'\left(\displaystyle{\frac{1}{2}}\right).
b) Explain what this number represents.
Determine the second derivative of each of the following:
y=xe^x-e^x
Determine the second derivative of
y=xe^{10x}
If y=\displaystyle{\frac{e^{2x}-1}{e^{2x}+1}}, \displaystyle{\frac{dy}{dx}}= ?
Determine the equation of the tangent to the curve defined by y=x-e^{-x} that is parallel to the line represented by 3x-y-9=0.
Determine the equation of the tangent to the curve y=x\sin x at the point where x=\displaystyle{\frac{\pi}{2}}.
An object moves along a line so that, at time t, its position is s=\dfrac{\sin t}{3+\cos2t}, where s is the displacement in metres. Calculate the object's velocity at t=\displaystyle{\frac{\pi}{4}}.
The number of bacteria in a culture, N, at time t is given by N(t)=2000[30+te^{-\displaystyle{\frac{t}{20}}}].
- When is the rate of change of the number of bacteria equal to zero?
The number of bacteria in a culture, N, at time t is given by N(t)=2000[30+te^{-\displaystyle{\frac{t}{20}}}].
- If the bacterial culture is placed in a colony of mice, the number of mice that become infected,
M, is related to the number of bacteria present by the equationM(t)=\sqrt[3]{N+1000}. After 10 days, how many mice are infected per day?
The concentrations of two medicines in the bloodstream t hours after injections are c_{1}(t)=te^{-1} and c_{2}(t)=t^2e^{-t}.
- Which medicine has the larger maximum concentration?
The concentrations of two medicines in the bloodstream t hours after injections are c_{1}(t)=te^{-1} and c_{2}(t)=t^2e^{-t}.
- Within the first half hour, which medicine has the larger maximum concentration?
Differentiate.
y=(2+3e^{-x})^3
Differentiate.
y=x^e
Differentiate.
y=e^{e^x}
Differentiate.
y=(1-e^{5x})^5
Differentiate.
y=5^x
Differentiate.
y=(0.47)^x
Differentiate.
y=(52)^{2x}
Differentiate.
y=5(2)^x
Differentiate.
y=4(e)^x
Differentiate.
y=-2(10)^{3x}
Determine y'.
y=\sin2^x
Determine y'.
y=x^2\sin x
Determine $y'$.
y=\sin \left( \displaystyle{\frac{\pi}{2}}-x \right)
Determine y'.
y=\cos x\sin x
Determine $y'$.
y=\cos^2x
Determine y'.
y=\cos x
Determine the equation of the tangent to the curve y=\cos x at \left(\displaystyle{\frac{\pi}{2}},0 \right) .
An object is suspended from the end of a spring. Its displacement from the equilibrium position is s=8\sin(10\pi t) at time t. Calculate the velocity and acceleration of the object at any time t, and show that \displaystyle{\frac{d^2s}{dt^2}}+100\pi^2s=0.
The position of a particle is given by s=5\cos\left(2t+\displaystyle{\frac{\pi}{4}}\right) at time t. What are the maximum values of the displacement, the velocity, and the acceleration?
The hypotenuse of a right triangle is 12 cm in length. Calculate the measure of the unknown angles in the triangle that will maximize its perimeter.
A fence is 1.5 m high and is 1 m from a wall. A ladder must start from the ground, touch the top of the fence, and rest somewhere on the wall. Calculate the minimum length of the ladder.
A thin rigid pole needs to be carried horizontally around a corner joining two corridors, which are 1 m and 0.8 m wide. Calculate the length of the longest pole that can be carried around this corner.
When the rules of hockey were developed, Canada did not use the metric system. Thus, the distance between the goal posts was designated to be six feed (slightly less than 2 m). If Sidney Crosby is on the goal line, three feet outside on of the goal posts, how far should he go out (perpendicular to the goal line) to maximize the angle in which he can shoot at the goal? Hint: Determine the values of x that maximize \theta in the following diagram.
Determine f''(x)
f(x)=4\sin^2(x-2)
Determine f''(x)
f(x)=2(\cos x)(\sec^2x)