Question 1

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

Accepted Answer

$$\cos^2$ x - \sin^2$ x$$

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 $$F(x) = f(g(x))$$, which formula correctly applies the chain rule?

Accepted Answer

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

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 is the correct application of the power rule combined with the chain rule for $$y = (x^2$ - 1$$)^{100}$$?

Accepted Answer

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

Question 6

If $$f(x) = e$$^{-2x + 5x}$$, what is f$$'(x)$$?

Accepted Answer

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

Question 7

Differentiate $$y = (2x + 1)^5$ (x^2$ - x^4$)^4$ using the product rule.

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 8

What is $$\frac{d}{dx} b^x$$ where $$b \u003e 0$$ and $$b 
eq 1$$?

Accepted Answer

$$b^x \ln b$$

Question 9

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

Accepted Answer

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

Question 10

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 11

If $$f(x) = e$$^{u(x)}$$, what is f$$'(x)$$?

Accepted Answer

$$e^{u(x)}$$ \cdot u'(x)$$

Question 12

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

Accepted Answer

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

Question 13

Which of the following best describes the chain rule?

Accepted Answer

It is used to differentiate composite functions.

Question 14

If $$g(x) = x^2$ \ln 2$$, what is $$g'(x)$$?

Accepted Answer

$$2x \ln 2$$

Question 15

Given $$f(x) = 4^x$$, what is $$f'(x)$$?

Accepted Answer

$$4^x \ln 4$$

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

Study Guide - Practice Questions