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 the curve $$y = \frac{1 + \sin x}{\cos x} at $$the point $$\left( \pi, -1 \right).$$

Accepted Answer

$$y + 1 = (1)(x - \pi)$$

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

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

Question 4

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

Accepted Answer

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

Question 5

Which of the following correctly applies the power rule combined with the chain rule to $$y = (x^2$ - 1$$)^{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

$$y' = 5(2x + 1)^4$ \cdot 2 (x^2$ - x^4$)^4$$ + $$4(x^2$$ - $$x^4)^3$ (2x + 1)^5$ (2x - 4x^3$)$$

Question 8

What is the derivative of $$y = b^x$$ with respect to $$x$$?

Accepted Answer

$$\frac{d}{dx} b^x = b^x \ln b$$

Question 9

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

Accepted Answer

$$\frac{dy}{dx} = -\frac{x}{y}$$

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)$$ in terms of u(x$$)$$?

Accepted Answer

$$f'(x) = e$$^{u(x)}$$ u'(x)$$

Question 12

Let $$y = e$$^{\sin x}$$. What is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 13

If $$f(x) = x^n$$, what is $$f'(x)$$?

Accepted Answer

$$n x$$^{n-1}$$

Question 14

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

Accepted Answer

$$2^x (\ln 2)^2$

Question 15

Which of the following is the correct application of the chain rule to $$y = (3x^2$ + 2)^7$?

Accepted Answer

$$y' = 7(3x^2$ + 2)^6$ \cdot 6x$$

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

Study Guide - Practice Questions