# 3.9 E: Derivatives Ln, etc. Exercises

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## 3.9: Derivatives of Ln, General Exponent & Log Functions; and Logarithmic Differentiation

### Exercise:

For the following exercises, find $$f′(x)$$ for each function.

333) $$f(x)=e^{x^3\ln x}$$

$$e^{x^3}\ln x(3x^2\ln x+x^2)$$

336) $$f(x)=\frac{10^x}{\ln10}$$

337) $$f(x)=2^{4x}+4x^2$$

$$2^{4x+2}⋅\ln 2+8x$$

338) $$f(x)=3^{\sin(3x)}$$

339) $$f(x)=x^π⋅π^x$$

$$πx^{π−1}⋅π^x+x^π⋅π^x\ln π$$

340) $$f(x)=\ln(4x^3+x)$$

341) $$f(x)=\ln\sqrt{5x−7}$$

$$\frac{5}{2(5x−7)}$$

342) $$f(x)=x^2\ln(9x)$$

343) $$f(x)=log(\sec x)$$

$$\frac{\tan x}{\ln10}$$

344) $$f(x)=log_7(6x^4+3)^5$$

345) $$f(x)=2^x⋅log_37^{x^2−4}$$

$$2^x⋅\ln 2⋅log_37^{x^2−4}+2^x⋅\frac{2x\ln 7}{\ln 3}$$

For the following exercises, use logarithmic differentiation to find $$\frac{dy}{dx}$$.

346) $$y=x^{\sqrt{x}}$$

347) $$y=(\sin(2x))^{4x}$$

$$(\sin(2x))^{4x}[4⋅ln(\sin(2x))+8x⋅\cot(2x)]$$

348) $$y=(\ln x)^{\ln x}$$

349) $$y=x^{log_2x}$$

$$x^{log_2x}⋅\frac{2\ln x}{x\ln 2}$$

350) $$y=(x^2−1)^{\ln x}$$

351) $$y=x^{\cot x}$$

$$x^{\cot x}⋅[−\csc^2x⋅lnx+\frac{\cot x}{x}]$$

352) $$y=\frac{x+11}{\sqrt[3]{x^2−4}}$$

353) $$y=x^{−1/2}(x^2+3)^{2/3}(3x−4)^4$$

$$x^{−1/2}(x^2+3)^{2/3}(3x−4)^4⋅[\frac{−1}{2x}+\frac{4x}{3(x^2+3)}+\frac{12}{3x−4}]$$

354) [T] Find an equation of the tangent line to the graph of $$f(x)=4xe^{(x^2−1)}$$ at the point where

$$x=−1.$$ Graph both the function and the tangent line.

355) [T] Find the equation of the line that is normal to the graph of $$f(x)=x⋅5^x$$ at the point where $$x=1$$. Graph both the function and the normal line.

$$y=\frac{−1}{5+5\ln 5}x+(5+\frac{1}{5+5\ln 5})$$

356) [T] Find the equation of the tangent line to the graph of $$x^3−x\ln y+y^3=2x+5$$ at the point where $$x=2$$. (Hint: Use implicit differentiation to find $$\frac{dy}{dx}$$.) Graph both the curve and the tangent line.

J357)

use the graph of $$y=f(x)$$ (shown below) to

a. sketch the graph of $$y=f^{−1}(x)$$, and

b. use part a. to estimate $$(f^{−1})′(1)$$.

a.

b. $$(f^{−1})′(1)~2$$

For the next set of exercises, find $$\frac{dy}{dx}$$. [Hint: first take the ln of both sides.]

J358) $$y=\frac{(2x^3−15x)\sqrt{6x^{4}+7}}{3x^2−x+3}$$

J359) $$y={30x^4}\sqrt{17x+2}{(\sin(x))}$$

J360) $$y={e^{5x}}{(3x-1)^\frac{2}{3}}{(8^{3x})}$$

3.9 E: Derivatives Ln, etc. Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.