\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus
Please login to create an exam or a quiz.
Use implicit differentiation where \(\displaystyle y\) is a function of \(\displaystyle x\) to find the derivative of \(\displaystyle - 4 e^{x^{3}} \log{\left(y \right)} - 4 \sin{\left(x^{3} \right)} \sin{\left(y^{3} \right)}=-44\)
Taking the derivative of both sides using implicit differentiation gives: \begin{equation*} - 12 x^{2} e^{x^{3}} \log{\left(y \right)} - 12 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} - 12 y^{2} y' \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} - \frac{4 y' e^{x^{3}}}{y} = 0 \end{equation*} Solving for \(\displaystyle y'\) gives \(\displaystyle y' = - \frac{3 x^{2} y \left(e^{x^{3}} \log{\left(y \right)} + \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)}\right)}{3 y^{3} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + e^{x^{3}}}\)
\begin{question}Use implicit differentiation where $y$ is a function of $x$ to find the derivative of $- 4 e^{x^{3}} \log{\left(y \right)} - 4 \sin{\left(x^{3} \right)} \sin{\left(y^{3} \right)}=-44$
\soln{9cm}{Taking the derivative of both sides using implicit differentiation gives:
\begin{equation*} - 12 x^{2} e^{x^{3}} \log{\left(y \right)} - 12 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} - 12 y^{2} y' \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} - \frac{4 y' e^{x^{3}}}{y} = 0 \end{equation*}
Solving for $y'$ gives $y' = - \frac{3 x^{2} y \left(e^{x^{3}} \log{\left(y \right)} + \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)}\right)}{3 y^{3} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + e^{x^{3}}}$}
\end{question}
\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}<p> <p>Use implicit differentiation where <img class="equation_image" title=" \displaystyle y " src="/equation_images/%20%5Cdisplaystyle%20y%20" alt="LaTeX: \displaystyle y " data-equation-content=" \displaystyle y " /> is a function of <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> to find the derivative of <img class="equation_image" title=" \displaystyle - 4 e^{x^{3}} \log{\left(y \right)} - 4 \sin{\left(x^{3} \right)} \sin{\left(y^{3} \right)}=-44 " src="/equation_images/%20%5Cdisplaystyle%20-%204%20e%5E%7Bx%5E%7B3%7D%7D%20%5Clog%7B%5Cleft%28y%20%5Cright%29%7D%20-%204%20%5Csin%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%20%5Csin%7B%5Cleft%28y%5E%7B3%7D%20%5Cright%29%7D%3D-44%20" alt="LaTeX: \displaystyle - 4 e^{x^{3}} \log{\left(y \right)} - 4 \sin{\left(x^{3} \right)} \sin{\left(y^{3} \right)}=-44 " data-equation-content=" \displaystyle - 4 e^{x^{3}} \log{\left(y \right)} - 4 \sin{\left(x^{3} \right)} \sin{\left(y^{3} \right)}=-44 " /> </p> </p><p> <p>Taking the derivative of both sides using implicit differentiation gives:
<img class="equation_image" title=" - 12 x^{2} e^{x^{3}} \log{\left(y \right)} - 12 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} - 12 y^{2} y' \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} - \frac{4 y' e^{x^{3}}}{y} = 0 " src="/equation_images/%20%20-%2012%20x%5E%7B2%7D%20e%5E%7Bx%5E%7B3%7D%7D%20%5Clog%7B%5Cleft%28y%20%5Cright%29%7D%20-%2012%20x%5E%7B2%7D%20%5Csin%7B%5Cleft%28y%5E%7B3%7D%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%20-%2012%20y%5E%7B2%7D%20y%27%20%5Csin%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28y%5E%7B3%7D%20%5Cright%29%7D%20-%20%5Cfrac%7B4%20y%27%20e%5E%7Bx%5E%7B3%7D%7D%7D%7By%7D%20%3D%200%20%20" alt="LaTeX: - 12 x^{2} e^{x^{3}} \log{\left(y \right)} - 12 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} - 12 y^{2} y' \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} - \frac{4 y' e^{x^{3}}}{y} = 0 " data-equation-content=" - 12 x^{2} e^{x^{3}} \log{\left(y \right)} - 12 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} - 12 y^{2} y' \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} - \frac{4 y' e^{x^{3}}}{y} = 0 " />
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{3 x^{2} y \left(e^{x^{3}} \log{\left(y \right)} + \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)}\right)}{3 y^{3} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + e^{x^{3}}} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20-%20%5Cfrac%7B3%20x%5E%7B2%7D%20y%20%5Cleft%28e%5E%7Bx%5E%7B3%7D%7D%20%5Clog%7B%5Cleft%28y%20%5Cright%29%7D%20%2B%20%5Csin%7B%5Cleft%28y%5E%7B3%7D%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%5Cright%29%7D%7B3%20y%5E%7B3%7D%20%5Csin%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28y%5E%7B3%7D%20%5Cright%29%7D%20%2B%20e%5E%7Bx%5E%7B3%7D%7D%7D%20" alt="LaTeX: \displaystyle y' = - \frac{3 x^{2} y \left(e^{x^{3}} \log{\left(y \right)} + \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)}\right)}{3 y^{3} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + e^{x^{3}}} " data-equation-content=" \displaystyle y' = - \frac{3 x^{2} y \left(e^{x^{3}} \log{\left(y \right)} + \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)}\right)}{3 y^{3} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + e^{x^{3}}} " /> </p> </p>