Question 1

Which of the following best describes the meaning of $$\frac{dy}{dx} at a $$point $$x$$ for a function $$y = f(x)$$?

Accepted Answer

It is the slope of the tangent line to the graph of $$y at$$ $$x.$$

Question 2

Given $$f(x) = 3x^4$ - 2x^2$ + 5$$, what is $$\frac{df}{dx}$$?

Accepted Answer

$$12x^3 - 4x$

Question 3

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

Accepted Answer

$$\cos(x) \cdot e^x + \sin(x) \cdot e^x$$

Question 4

Which of the following is the correct application of the quotient rule for $$y = \frac{f(x)}{g(x)}$$?

Accepted Answer

$$\frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$

Question 5

A biologist models the population of bacteria as $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of change of the population at time t$?

Accepted Answer

$$0.05 \cdot 100e$$^{0.05t}$$

Question 6

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

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 7

Suppose $$y^2$$ + $$x^2 = 25$. Using implicit differentiation, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 8

Which of the following is the derivative of $$	anh(x)$$?

Accepted Answer

$$	ext{sech}^2(x)$$

Question 9

A chemical reaction's rate is modeled by $$R(t) = \frac{e$$^{2t}$$}{t^2$ + 1}. $$What is $$\frac{dR}{dt}$$?

Accepted Answer

$$\frac{2e$$^{2t}$$(t^2 + 1) - 2t$$ $$e^{2t}$$}{$$(t^2$$ + $$1)^2$$}$

Question 10

If $$y = \arcsinh(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{\sqrt{x^2$ + 1}}$$

Question 11

Which of the following is the correct derivative of $$y = \cosh(x)$$?

Accepted Answer

$$\sinh(x)$$

Question 12

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

Accepted Answer

$$10x(x^2$$ + $$1)^4$$

Question 13

A company's profit is given by $$P(x) = x \ln(x)$$, where $$x is $$the number of units sold. What is $$\frac{dP}{dx}$$?

Accepted Answer

$$\ln(x) + 1$$

Question 14

Which of the following is the correct derivative of $$y = \arctanh(x)$$?

Accepted Answer

$$\frac{1}{1 - x^2$}$$

Question 15

If $$y = \ln(\sin(x))$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\cot(x)$$

Question 16

A medication's concentration in the bloodstream is modeled by $$C(t) = \ln(2t + 1). $$What is the rate of change of concentration with respect to time?

Accepted Answer

$$\frac{2}{2t + 1}$$

Question 17

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

Accepted Answer

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

Question 18

Which of the following is the correct derivative of $$y = 	ext{sech}(x)$$?

Accepted Answer

$$-	ext{sech}(x)	anh(x)$$

Question 19

If $$y = \arccosh(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{\sqrt{x^2$ - 1}}$$

Question 20

A physics experiment measures voltage as $$V(t) = \cosh(3t). $$What is $$\frac{dV}{dt}$$?

Accepted Answer

$$3\sinh(3t)$$

Differentiation Techniques: Notation, Rules, and Special Functions

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