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 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 $$F(x) = f(g(x))$$, which of the following correctly expresses $$F'(x)$$ using the chain rule?

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

Which of the following is the correct application of the power rule combined with the chain rule for $$\frac{d}{dx} [U^n]$$?

Accepted Answer

$$nU^{n-1}$$ \frac{dU}{dx}$$

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

Differentiate $$y = (2x + 1)^5$ (x^2$ + x^4$)^{10}$.

Accepted Answer

$$5(2x + 1)^4$ \cdot 2 (x^2$ + x^4$)^{10}$$ + $$10(x^2$$ + $$x^4)^9$ (2x + 1)^5$ (2x + 4x^3$)$$

Question 9

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

Accepted Answer

$$b^x \ln b$$

Question 10

If $$x^2$$ + $$y^2 = 25$, find $$\frac{dy}{dx}$$.

Accepted Answer

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

Question 11

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 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

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

Techniques of Differentiation: The Chain Rule, Power Rule, and Applications

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