Question 1

What is the derivative of $$\sin u$$ with respect to $$x$$, where $$u is a $$differentiable function of $$x$$?

Accepted Answer

$$\cos u \cdot u'$$

Question 2

Which of the following is the correct derivative of $$\ln u$$ with respect to $$x$$, where $$u is a $$differentiable function of $$x$$?

Accepted Answer

$$\frac{1}{u} \cdot u'$$

Question 3

Find the derivative of $$	an$$^{-1}$$ u$$ with respect to $$x$$, where $$u is a $$differentiable function of $$x.$$

Accepted Answer

$$\frac{1}{1+u^2$} \cdot u'$$

Question 4

A medication's concentration in the bloodstream is modeled by $$C(t) = e$$^{kt}$$, where k$ is a constant. What is $$\frac{dC}{dt}$$?

Accepted Answer

$$k e$$^{kt}$$

Question 5

What is the derivative of $$\cos$$^{-1}$$ u$$ with respect to $$x$$, where $$u is a $$differentiable function of $$x$$?

Accepted Answer

$$-\frac{1}{\sqrt{1-u^2$}} \cdot u'$$

Question 6

If $$y = \log_a u$$, where $$u is a $$differentiable function of $$x$$ and $$a \u003e 0$$, $$a 
eq 1$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{u \ln a} \cdot u'$$

Question 7

A population grows according to $$P(t) = a^t$$, where $$a \u003e 0. $$What is $$\frac{dP}{dt}$$?

Accepted Answer

$$a^t \ln a$$

Question 8

Find the derivative of $$\sec u$$ with respect to $$x$$, where $$u is a $$differentiable function of $$x.$$

Accepted Answer

$$\sec u 	an u \cdot u'$$

Question 9

If $$y = \sin$$^{-1}$$ u$$, where $$u is a $$differentiable function of $$x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{\sqrt{1-u^2$}} \cdot u'$$

Question 10

A chemical reaction rate is given by $$R(t) = \ln(u(t))$$, where $$u(t) is $$the concentration of a reactant. What is $$\frac{dR}{dt}$$?

Accepted Answer

$$\frac{1}{u(t)} \cdot u'(t)$$

Question 11

Find the derivative of $$\cot u$$ with respect to $$x$$, where $$u is a $$differentiable function of $$x.$$

Accepted Answer

$$-\csc^2$ u \cdot u'$$

Question 12

If $$y = f$$^{-1}$$(x)$$ and $$f is $$differentiable and invertible, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{f'(y)}$$

Question 13

Find the derivative of $$\csc u$$ with respect to $$x$$, where $$u is a $$differentiable function of $$x.$$

Accepted Answer

$$-\csc u \cot u \cdot u'$$

Question 14

If $$y = 	an u$$, where $$u is a $$differentiable function of $$x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\sec^2$ u \cdot u'$$

Question 15

A function $$y = f(g(x))$$ models the temperature in a chemical process, where $$g(x) is $$the time-dependent input. What is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 16

Find the derivative of $$\sec$$^{-1}$$ u$$ with respect to $$x$$, where $$u is a $$differentiable function of $$x.$$

Accepted Answer

$$\frac{1}{|u| \sqrt{u^2$-1}} \cdot u'$$

Question 17

If $$y = \cot$$^{-1}$$ u$$, where $$u is a $$differentiable function of $$x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{1+u^2$} \cdot u'$$

Question 18

Given $$y = u(x) v(x)$$, where $$u$$ and $$v$$ are differentiable functions of $$x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$u' v + u v'$$

Question 19

If $$y = e$$^{u(x)}$$, where u$ is a differentiable function of x$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 20

A physics experiment measures $$y = \csc$$^{-1}$$ u$$, where $$u is a $$differentiable function of $$x. $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$-\frac{1}{|u| \sqrt{u^2$-1}} \cdot u'$$

Mastering Derivatives: Calculus Study Guide

Study Guide - Practice Questions