$$y'' = \frac{x^2$(1-2\tan x) - 2x\sec^2$ x}{x^4$}$$

$$y' = \frac{2x+1}{x^2$+x+1} - 2\tan x - 1$$

$$y' = \ln(\cos x) + x \cdot (-\tan x)$$

Question 1

Let $$y = rac{	an x}{x^2$}. $$What is the second derivative $$y''$$ with respect to $$x$$?

Accepted Answer

$$y'' = \frac{x^2$(1-2	an x) - 2x\sec^2$ x}{x^4$}$$

Question 2

Given $$y = \log_2$(e^{-x}$$ \cos(nx))$$, what is y$$'$$ with respect to x$?

Accepted Answer

$$y' = \frac{-e$$^{-x}$$\cos(nx) + e$$^{-x}$$(-\sin(nx) n)}{e$$^{-x}$$\cos(nx) \ln 2}$$

Question 3

For $$y = (\sin x$$)^{\ln x}$$, what is y$$'$$?

Accepted Answer

$$y' = \frac{\cot x}{x} (\sin x$$)^{\ln x}$$

Question 4

Let $$y = \ln\left(\frac{e$$^{-x}$$\cos^2$ x}{x^2$ + x + 1}\right). $$What is $$y'$$?

Accepted Answer

$$y' = \frac{2x+1}{x^2$+x+1} - 2	an x - 1$$

Question 5

If $$y = \ln x$$, what is $$y''$$?

Accepted Answer

$$y'' = -\frac{1}{x^2$}$$

Question 6

A medication's concentration in the bloodstream is modeled by $$C(t) = e$$^{-kt}$$ \cos(at)$$, where $$k, a \u003e 0. $$What is $$\frac{dC}{dt}$$?

Accepted Answer

$$\frac{dC}{dt} = -k e$$^{-kt}$$ \cos(at) - a e$$^{-kt}$$ \sin(at)$$

Question 7

Suppose $$y = x \ln(\cos x). $$What is $$y'$$?

Accepted Answer

$$y' = \ln(\cos x) + x \cdot (-	an x)$$

Question 8

Given $$y = \log_2$(e^{-x}$$ \cos(nx))$$, which of the following best describes the process to find y$$'$?

Accepted Answer

Apply the chain rule to the logarithm, then differentiate the inner function using the product and chain rules.

Question 9

If $$y = x + a \ln(\cos x)$$, what is $$y'$$?

Accepted Answer

$$y' = 1 - a 	an x$$

Question 10

A population grows according to $$P(t) = e$$^{rt}$$ \cos(bt)$$, where $$r, b \u003e 0. $$What is $$\frac{dP}{dt}$$?

Accepted Answer

$$\frac{dP}{dt} = r e$$^{rt}$$ \cos(bt) - b e$$^{rt}$$ \sin(bt)$$

Question 11

Let $$y = \ln(x^2$ + x + 1). $$What is $$y'$$?

Accepted Answer

$$y' = \frac{2x+1}{x^2$ + x + 1}$$

Question 12

If $$y = \ln(e$$^{-x}$$ \cos^2$ x)$$, which of the following is the correct simplification for $$y$$?

Accepted Answer

$$y = -x + 2 \ln(\cos x)$$

Question 13

Given $$y = (\sin x$$)^{\ln x}$$, which of the following is the correct logarithmic form for y$?

Accepted Answer

$$\ln y = \ln x \ln(\sin x)$$

Question 14

If $$y = \ln(x^2$ + x + 1)$$, what is $$y''$$?

Accepted Answer

$$y'' = \frac{2(x^2$ + x + 1) - (2x+1)^2$}{(x^2$ + x + 1)^2$}$$

Question 15

A chemical reaction rate is given by $$R(t) = e$$^{-kt}$$ \cos^2$(at). $$What is $$\frac{dR}{dt}$$?

Accepted Answer

$$\frac{dR}{dt} = -k e$$^{-kt}$$ \cos^2$(at) - 2a e$$^{-kt}$$ \cos(at) \sin(at)$$

Question 16

If $$y = \ln(\sin x$$)^{\ln x}$$, what is y$$'$$?

Accepted Answer

$$y' = \frac{\cot x}{x} (\sin x$$)^{\ln x}$$

Question 17

Let $$y = x \ln x. $$What is $$y'$$?

Accepted Answer

$$y' = \ln x + 1$$

Question 18

Suppose $$y = \ln(e$$^{-x}$$ \cos^2$ x). $$What is $$y'$$?

Accepted Answer

$$y' = -1 - 2 	an x$$

Advanced Differentiation Techniques in Calculus: Logarithmic and Implicit Differentiation

Study Guide - Practice Questions