\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus

Please login to create an exam or a quiz.

Calculus
Derivatives
New Random

Find the derivative of \(\displaystyle y=\sqrt[4]{\ln{\left(x \right)}}\).

The outer function is \(\displaystyle f(u) = \sqrt[4]{u}\) and the inner function \(\displaystyle u = \ln{\left(x \right)}\). The chain rule gives \(\displaystyle y'= \frac{df}{du}\frac{du}{dx}\). \(\displaystyle y' = \frac{1}{4 u^{\frac{3}{4}}}(\frac{1}{x}) = \frac{1}{4 x \ln{\left(x \right)}^{\frac{3}{4}}}\).

Download \(\LaTeX\) 

\begin{question}Find the derivative of $y=\sqrt[4]{\ln{\left(x \right)}}$. 
    \soln{9cm}{The outer function is $f(u) = \sqrt[4]{u}$ and the inner function $u = \ln{\left(x \right)}$. The chain rule gives $y'= \frac{df}{du}\frac{du}{dx}$. $y' = \frac{1}{4 u^{\frac{3}{4}}}(\frac{1}{x}) = \frac{1}{4 x \ln{\left(x \right)}^{\frac{3}{4}}}$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}

\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}

\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}

\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue

\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}
HTML for Canvas 
<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y=\sqrt[4]{\ln{\left(x \right)}} " src="/equation_images/%20%5Cdisplaystyle%20y%3D%5Csqrt%5B4%5D%7B%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%7D%20" alt="LaTeX:  \displaystyle y=\sqrt[4]{\ln{\left(x \right)}} " data-equation-content=" \displaystyle y=\sqrt[4]{\ln{\left(x \right)}} " /> . </p> </p>
HTML for Canvas
<p> <p>The outer function is  <img class="equation_image" title=" \displaystyle f(u) = \sqrt[4]{u} " src="/equation_images/%20%5Cdisplaystyle%20f%28u%29%20%3D%20%5Csqrt%5B4%5D%7Bu%7D%20" alt="LaTeX:  \displaystyle f(u) = \sqrt[4]{u} " data-equation-content=" \displaystyle f(u) = \sqrt[4]{u} " />  and the inner function  <img class="equation_image" title=" \displaystyle u = \ln{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20u%20%3D%20%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle u = \ln{\left(x \right)} " data-equation-content=" \displaystyle u = \ln{\left(x \right)} " /> . The chain rule gives  <img class="equation_image" title=" \displaystyle y'= \frac{df}{du}\frac{du}{dx} " src="/equation_images/%20%5Cdisplaystyle%20y%27%3D%20%5Cfrac%7Bdf%7D%7Bdu%7D%5Cfrac%7Bdu%7D%7Bdx%7D%20" alt="LaTeX:  \displaystyle y'= \frac{df}{du}\frac{du}{dx} " data-equation-content=" \displaystyle y'= \frac{df}{du}\frac{du}{dx} " /> .  <img class="equation_image" title=" \displaystyle y' = \frac{1}{4 u^{\frac{3}{4}}}(\frac{1}{x}) = \frac{1}{4 x \ln{\left(x \right)}^{\frac{3}{4}}} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B1%7D%7B4%20u%5E%7B%5Cfrac%7B3%7D%7B4%7D%7D%7D%28%5Cfrac%7B1%7D%7Bx%7D%29%20%3D%20%5Cfrac%7B1%7D%7B4%20x%20%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%5E%7B%5Cfrac%7B3%7D%7B4%7D%7D%7D%20" alt="LaTeX:  \displaystyle y' = \frac{1}{4 u^{\frac{3}{4}}}(\frac{1}{x}) = \frac{1}{4 x \ln{\left(x \right)}^{\frac{3}{4}}} " data-equation-content=" \displaystyle y' = \frac{1}{4 u^{\frac{3}{4}}}(\frac{1}{x}) = \frac{1}{4 x \ln{\left(x \right)}^{\frac{3}{4}}} " /> . </p> </p>