Question 1

Given $$y = \sin x \cos x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\cos x \cos x - \sin x \sin x$$

Question 2

Find the equation of the tangent line to $$y = \frac{1 + \sin x}{\cos x} at$$ $$x = \pi.$$

Accepted Answer

$$y = x - \pi - 1$$

Question 3

If $$F(x) = f(g(x))$$ and both $$f$$ and $$g$$ are differentiable, what is the derivative $$F'(x)$$ according to the chain rule?

Accepted Answer

$$f'(g(x)) \cdot g'(x)$$

Question 4

Differentiate $$F(x) = \$$left(2x^2$$ + 1\$$right)^{100}$$.

Accepted Answer

$$100 \$$left(2x^2$$ + 1\$$right)^{99}$$ \cdot 4x$$

Question 5

Which of the following correctly applies the chain rule to $$y = e$$^{\sin x}$$?

Accepted Answer

$$e^{\sin x}$$ \cos x$$

Question 6

If $$y = (2x+1)^5$ (x^2$+x^4)^4$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$5(2x+1)^4$$ \cdot 2 $$(x^2+x^4$)^4$$ + $$4(x^2+x^4$)^3$$ $$(2x+1)^5$$ $$(2x+4x^3$$)$$

Question 7

Given $$x^2$$ + $$y^2 = 25$, find $$\frac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

$$-\frac{x}{y}$$

Question 8

Find the equation of the tangent line to the circle $$x^2$$ + $$y^2 = 25$ at the point (3$$,4)$$.

Accepted Answer

$$y - 4 = -\frac{3}{4}(x - 3)$$

Question 9

Differentiate $$f(x) = e$$^{-2x+5x}$$.

Accepted Answer

$$(-2x+5) e$$^{-2x+5x}$$

Question 10

If $$y = b^x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$b^x \ln b$$

Question 11

Let $$f(x) = x^n$$ and $$g(x) = x^2$ \ln 2. $$What is $$f'(g(x))$$?

Accepted Answer

$$2x \ln 2 \cdot n (x^2$ \ln 2$$)^{n-1}$$

Question 12

Which of the following best describes the chain rule?

Accepted Answer

It is used to differentiate composite functions.

Question 13

If $$y = e$$^{x^2}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2x e$$^{x^2}$$

Question 14

Differentiate $$y = (x^2$ - 1$$)^{100}$$.

Accepted Answer

$$100(x^2$$ - $$1)^{99}$$ \cdot 2x$$

Question 15

If $$y = e$$^{\sin x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$e^{\sin x}$$ \cos x$$

Techniques of Differentiation: Chain Rule, Power Rule, and Tangent Lines

Study Guide - Practice Questions