\(\text{www.the}\beta\text{etafunction.com}\)
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 \(\displaystyle - 3 e^{y} \cos{\left(x \right)} - 2 \sin{\left(x^{3} \right)} \sin{\left(y^{3} \right)}=-41\)
Taking the derivative of both sides using implicit differentiation gives: \begin{equation*} - 6 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} - 6 y^{2} y' \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} - 3 y' e^{y} \cos{\left(x \right)} + 3 e^{y} \sin{\left(x \right)} = 0 \end{equation*} Solving for \(\displaystyle y'\) gives \(\displaystyle y' = \frac{- 2 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} + e^{y} \sin{\left(x \right)}}{2 y^{2} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + e^{y} \cos{\left(x \right)}}\)
\begin{question}Use implicit differentiation where $y$ is a function of $x$ to find the derivative of $- 3 e^{y} \cos{\left(x \right)} - 2 \sin{\left(x^{3} \right)} \sin{\left(y^{3} \right)}=-41$
\soln{9cm}{Taking the derivative of both sides using implicit differentiation gives:
\begin{equation*} - 6 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} - 6 y^{2} y' \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} - 3 y' e^{y} \cos{\left(x \right)} + 3 e^{y} \sin{\left(x \right)} = 0 \end{equation*}
Solving for $y'$ gives $y' = \frac{- 2 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} + e^{y} \sin{\left(x \right)}}{2 y^{2} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + e^{y} \cos{\left(x \right)}}$}
\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 - 3 e^{y} \cos{\left(x \right)} - 2 \sin{\left(x^{3} \right)} \sin{\left(y^{3} \right)}=-41 " src="/equation_images/%20%5Cdisplaystyle%20-%203%20e%5E%7By%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20-%202%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-41%20" alt="LaTeX: \displaystyle - 3 e^{y} \cos{\left(x \right)} - 2 \sin{\left(x^{3} \right)} \sin{\left(y^{3} \right)}=-41 " data-equation-content=" \displaystyle - 3 e^{y} \cos{\left(x \right)} - 2 \sin{\left(x^{3} \right)} \sin{\left(y^{3} \right)}=-41 " /> </p> </p><p> <p>Taking the derivative of both sides using implicit differentiation gives:
<img class="equation_image" title=" - 6 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} - 6 y^{2} y' \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} - 3 y' e^{y} \cos{\left(x \right)} + 3 e^{y} \sin{\left(x \right)} = 0 " src="/equation_images/%20%20-%206%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-%206%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-%203%20y%27%20e%5E%7By%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%203%20e%5E%7By%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%3D%200%20%20" alt="LaTeX: - 6 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} - 6 y^{2} y' \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} - 3 y' e^{y} \cos{\left(x \right)} + 3 e^{y} \sin{\left(x \right)} = 0 " data-equation-content=" - 6 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} - 6 y^{2} y' \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} - 3 y' e^{y} \cos{\left(x \right)} + 3 e^{y} \sin{\left(x \right)} = 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{- 2 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} + e^{y} \sin{\left(x \right)}}{2 y^{2} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + e^{y} \cos{\left(x \right)}} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B-%202%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%2B%20e%5E%7By%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B2%20y%5E%7B2%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%7By%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%7D%20" alt="LaTeX: \displaystyle y' = \frac{- 2 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} + e^{y} \sin{\left(x \right)}}{2 y^{2} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + e^{y} \cos{\left(x \right)}} " data-equation-content=" \displaystyle y' = \frac{- 2 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} + e^{y} \sin{\left(x \right)}}{2 y^{2} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + e^{y} \cos{\left(x \right)}} " /> </p> </p>