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

Please login to create an exam or a quiz.

Calculus
Derivatives
New Random

Find the derivative of \(\displaystyle f(x) = \tan^{-1}{\left(7 x \right)}\).

Taking the tangent of both sides gives \(\displaystyle \tan(y) = 7 x\). Taking the implicit derivative gives \(\displaystyle \sec^2(y)y' = 7\). Solving for \(\displaystyle y'\) gives \(\displaystyle y' = \frac{7}{\sec^{2}{\left(y \right)}}\). Using the trigonometric identity \(\displaystyle \tan^2(y)+1 = \sec^2(y)\) gives \(\displaystyle y' = \frac{7}{\tan^{2}{\left(y \right)} + 1}\). Using the fact that \(\displaystyle \tan(y)= 7 x\) gives \(\displaystyle y' = \frac{7}{49 x^{2} + 1}\). Note that the formula for the derivative + the chain rule could have also been used.

Download \(\LaTeX\) 

\begin{question}Find the derivative of $f(x) = \tan^{-1}{\left(7 x \right)}$. 
    \soln{9cm}{Taking the tangent of both sides gives $\tan(y) = 7 x$. Taking the implicit derivative gives $\sec^2(y)y' = 7$. Solving for $y'$ gives $y' = \frac{7}{\sec^{2}{\left(y \right)}}$. Using the trigonometric identity $\tan^2(y)+1 = \sec^2(y)$ gives $y' = \frac{7}{\tan^{2}{\left(y \right)} + 1}$.  Using the fact that $\tan(y)= 7 x$ gives $y' = \frac{7}{49 x^{2} + 1}$. Note that the formula for the derivative + the chain rule could have also been used.}

\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 f(x) = \tan^{-1}{\left(7 x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20%5Ctan%5E%7B-1%7D%7B%5Cleft%287%20x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f(x) = \tan^{-1}{\left(7 x \right)} " data-equation-content=" \displaystyle f(x) = \tan^{-1}{\left(7 x \right)} " /> . </p> </p>
HTML for Canvas
<p> <p>Taking the tangent of both sides gives  <img class="equation_image" title=" \displaystyle \tan(y) = 7 x " src="/equation_images/%20%5Cdisplaystyle%20%5Ctan%28y%29%20%3D%207%20x%20" alt="LaTeX:  \displaystyle \tan(y) = 7 x " data-equation-content=" \displaystyle \tan(y) = 7 x " /> . Taking the implicit derivative gives  <img class="equation_image" title=" \displaystyle \sec^2(y)y' = 7 " src="/equation_images/%20%5Cdisplaystyle%20%5Csec%5E2%28y%29y%27%20%3D%207%20" alt="LaTeX:  \displaystyle \sec^2(y)y' = 7 " data-equation-content=" \displaystyle \sec^2(y)y' = 7 " /> . Solving for  <img class="equation_image" title=" \displaystyle y' " src="/equation_images/%20%5Cdisplaystyle%20y%27%20" alt="LaTeX:  \displaystyle y' " data-equation-content=" \displaystyle y' " />  gives  <img class="equation_image" title=" \displaystyle y' = \frac{7}{\sec^{2}{\left(y \right)}} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B7%7D%7B%5Csec%5E%7B2%7D%7B%5Cleft%28y%20%5Cright%29%7D%7D%20" alt="LaTeX:  \displaystyle y' = \frac{7}{\sec^{2}{\left(y \right)}} " data-equation-content=" \displaystyle y' = \frac{7}{\sec^{2}{\left(y \right)}} " /> . Using the trigonometric identity  <img class="equation_image" title=" \displaystyle \tan^2(y)+1 = \sec^2(y) " src="/equation_images/%20%5Cdisplaystyle%20%5Ctan%5E2%28y%29%2B1%20%3D%20%5Csec%5E2%28y%29%20" alt="LaTeX:  \displaystyle \tan^2(y)+1 = \sec^2(y) " data-equation-content=" \displaystyle \tan^2(y)+1 = \sec^2(y) " />  gives  <img class="equation_image" title=" \displaystyle y' = \frac{7}{\tan^{2}{\left(y \right)} + 1} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B7%7D%7B%5Ctan%5E%7B2%7D%7B%5Cleft%28y%20%5Cright%29%7D%20%2B%201%7D%20" alt="LaTeX:  \displaystyle y' = \frac{7}{\tan^{2}{\left(y \right)} + 1} " data-equation-content=" \displaystyle y' = \frac{7}{\tan^{2}{\left(y \right)} + 1} " /> .  Using the fact that  <img class="equation_image" title=" \displaystyle \tan(y)= 7 x " src="/equation_images/%20%5Cdisplaystyle%20%5Ctan%28y%29%3D%207%20x%20" alt="LaTeX:  \displaystyle \tan(y)= 7 x " data-equation-content=" \displaystyle \tan(y)= 7 x " />  gives  <img class="equation_image" title=" \displaystyle y' = \frac{7}{49 x^{2} + 1} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B7%7D%7B49%20x%5E%7B2%7D%20%2B%201%7D%20" alt="LaTeX:  \displaystyle y' = \frac{7}{49 x^{2} + 1} " data-equation-content=" \displaystyle y' = \frac{7}{49 x^{2} + 1} " /> . Note that the formula for the derivative + the chain rule could have also been used.</p> </p>