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

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

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

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

Answer:: $2^{4x+2}⋅\ln 2+8x$

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

339) $f(x)=x^π⋅π^x$

Answer:: $πx^{π−1}⋅π^x+x^π⋅π^x\ln π$

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

341) $f(x)=\ln\sqrt{5x−7}$

Answer:: $\frac{5}{2(5x−7)}$

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

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

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

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

345) $f(x)=2^x⋅log_37^{x^2−4}$

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

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

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

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

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

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

349) $y=x^{log_2x}$

Answer:: $x^{log_2x}⋅\frac{2\ln x}{x\ln 2}$

350) $y=(x^2−1)^{\ln x}$

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

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

352) $y=\frac{x+11}{\sqrt[3]{x^2−4}}$

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

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

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.

Answer:

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

$CNX_Calc_Figure_03_09_202.jpeg$

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 $(f^{−1})′(1)$.

$CNX_Calc_Figure_03_07_203.jpeg$

Answer:

$CNX_Calc_Figure_03_07_204.jpeg$

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