3.9 E: Derivatives Ln, etc. Exercises

Last updated
Save as PDF

Page ID: 10870

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$ \newcommand{\dsum}{\displaystyle\sum\limits} $

$ \newcommand{\dint}{\displaystyle\int\limits} $

$ \newcommand{\dlim}{\displaystyle\lim\limits} $

$ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\id}{\mathrm{id}}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\kernel}{\mathrm{null}\,}$

$ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$

$ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$

$ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$\newcommand{\longvect}{\overrightarrow}$

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\ket}[1]{\left| #1 \right>}$
$\newcommand{\bra}[1]{\left< #1 \right|}$
$\newcommand{\braket}[2]{\left< #1 \vphantom{#2} \right| \left. #2 \vphantom{#1} \right>}$
$\newcommand{\braopket}[3]{\left< #1 \vphantom{#2}\vphantom{#3} \right| #2 \vphantom{#1}\vphantom{#3} \left| #3 \vphantom{#1}\vphantom{#2} \right>}$
$\newcommand{\qmvec}[1]{\mathbf{\vec{#1}}}$
$\newcommand{\op}[1]{\hat{\mathbf{#1}}}$
$\newcommand{\expect}[1]{\langle #1 \rangle}$
$\newcommand{\dfn}[1]{\emph{\textbf{#1}}}$

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

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

Search

Text Color

Text Size

Margin Size

Font Type