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

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

Accepted Answer

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

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

Given $$y = \log_a u$$, 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

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 8

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

Accepted Answer

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

Question 9

What is 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 10

Given $$y = a$$^{u(x)}$$ where a$$ \u003e 0$$ and a$$ 
eq 1$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$a^{u(x)}$$ \ln a \cdot u'(x)$$

Question 11

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 12

If $$y = \sin$$^{-1}$$ u$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

$$-\sin u \cdot u'$$

Question 15

Given $$y = \sec$$^{-1}$$ u$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 16

If $$y = \csc$$^{-1}$$ u$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

$$u'v + uv'$$

Question 19

Given $$y = \frac{u(x)}{v(x)}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{u'v - uv'}{v^2$}$$

Question 20

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

Accepted Answer

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

Step-by-Step Guidance for Derivative Formulas and Rules

Study Guide - Practice Questions