\(\text{www.the}\beta\text{etafunction.com}\)
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 \(\displaystyle 21 \sqrt{y} \cos{\left(x^{3} \right)} + 4 \log{\left(y \right)} \sin{\left(x^{2} \right)}=41\)
Taking the derivative of both sides using implicit differentiation gives: \begin{equation*} - 63 x^{2} \sqrt{y} \sin{\left(x^{3} \right)} + 8 x \log{\left(y \right)} \cos{\left(x^{2} \right)} + \frac{4 y' \sin{\left(x^{2} \right)}}{y} + \frac{21 y' \cos{\left(x^{3} \right)}}{2 \sqrt{y}} = 0 \end{equation*} Solving for \(\displaystyle y'\) gives \(\displaystyle y' = \frac{2 x \left(63 x y^{2} \sin{\left(x^{3} \right)} - 8 y^{\frac{3}{2}} \log{\left(y \right)} \cos{\left(x^{2} \right)}\right)}{8 \sqrt{y} \sin{\left(x^{2} \right)} + 21 y \cos{\left(x^{3} \right)}}\)
\begin{question}Use implicit differentiation where $y$ is a function of $x$ to find the derivative of $21 \sqrt{y} \cos{\left(x^{3} \right)} + 4 \log{\left(y \right)} \sin{\left(x^{2} \right)}=41$
\soln{9cm}{Taking the derivative of both sides using implicit differentiation gives:
\begin{equation*} - 63 x^{2} \sqrt{y} \sin{\left(x^{3} \right)} + 8 x \log{\left(y \right)} \cos{\left(x^{2} \right)} + \frac{4 y' \sin{\left(x^{2} \right)}}{y} + \frac{21 y' \cos{\left(x^{3} \right)}}{2 \sqrt{y}} = 0 \end{equation*}
Solving for $y'$ gives $y' = \frac{2 x \left(63 x y^{2} \sin{\left(x^{3} \right)} - 8 y^{\frac{3}{2}} \log{\left(y \right)} \cos{\left(x^{2} \right)}\right)}{8 \sqrt{y} \sin{\left(x^{2} \right)} + 21 y \cos{\left(x^{3} \right)}}$}
\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 21 \sqrt{y} \cos{\left(x^{3} \right)} + 4 \log{\left(y \right)} \sin{\left(x^{2} \right)}=41 " src="/equation_images/%20%5Cdisplaystyle%2021%20%5Csqrt%7By%7D%20%5Ccos%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%20%2B%204%20%5Clog%7B%5Cleft%28y%20%5Cright%29%7D%20%5Csin%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%3D41%20" alt="LaTeX: \displaystyle 21 \sqrt{y} \cos{\left(x^{3} \right)} + 4 \log{\left(y \right)} \sin{\left(x^{2} \right)}=41 " data-equation-content=" \displaystyle 21 \sqrt{y} \cos{\left(x^{3} \right)} + 4 \log{\left(y \right)} \sin{\left(x^{2} \right)}=41 " /> </p> </p><p> <p>Taking the derivative of both sides using implicit differentiation gives:
<img class="equation_image" title=" - 63 x^{2} \sqrt{y} \sin{\left(x^{3} \right)} + 8 x \log{\left(y \right)} \cos{\left(x^{2} \right)} + \frac{4 y' \sin{\left(x^{2} \right)}}{y} + \frac{21 y' \cos{\left(x^{3} \right)}}{2 \sqrt{y}} = 0 " src="/equation_images/%20%20-%2063%20x%5E%7B2%7D%20%5Csqrt%7By%7D%20%5Csin%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%20%2B%208%20x%20%5Clog%7B%5Cleft%28y%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%20%2B%20%5Cfrac%7B4%20y%27%20%5Csin%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%7D%7By%7D%20%2B%20%5Cfrac%7B21%20y%27%20%5Ccos%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%7D%7B2%20%5Csqrt%7By%7D%7D%20%3D%200%20%20" alt="LaTeX: - 63 x^{2} \sqrt{y} \sin{\left(x^{3} \right)} + 8 x \log{\left(y \right)} \cos{\left(x^{2} \right)} + \frac{4 y' \sin{\left(x^{2} \right)}}{y} + \frac{21 y' \cos{\left(x^{3} \right)}}{2 \sqrt{y}} = 0 " data-equation-content=" - 63 x^{2} \sqrt{y} \sin{\left(x^{3} \right)} + 8 x \log{\left(y \right)} \cos{\left(x^{2} \right)} + \frac{4 y' \sin{\left(x^{2} \right)}}{y} + \frac{21 y' \cos{\left(x^{3} \right)}}{2 \sqrt{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{2 x \left(63 x y^{2} \sin{\left(x^{3} \right)} - 8 y^{\frac{3}{2}} \log{\left(y \right)} \cos{\left(x^{2} \right)}\right)}{8 \sqrt{y} \sin{\left(x^{2} \right)} + 21 y \cos{\left(x^{3} \right)}} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B2%20x%20%5Cleft%2863%20x%20y%5E%7B2%7D%20%5Csin%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%20-%208%20y%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%20%5Clog%7B%5Cleft%28y%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%5Cright%29%7D%7B8%20%5Csqrt%7By%7D%20%5Csin%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%20%2B%2021%20y%20%5Ccos%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%7D%20" alt="LaTeX: \displaystyle y' = \frac{2 x \left(63 x y^{2} \sin{\left(x^{3} \right)} - 8 y^{\frac{3}{2}} \log{\left(y \right)} \cos{\left(x^{2} \right)}\right)}{8 \sqrt{y} \sin{\left(x^{2} \right)} + 21 y \cos{\left(x^{3} \right)}} " data-equation-content=" \displaystyle y' = \frac{2 x \left(63 x y^{2} \sin{\left(x^{3} \right)} - 8 y^{\frac{3}{2}} \log{\left(y \right)} \cos{\left(x^{2} \right)}\right)}{8 \sqrt{y} \sin{\left(x^{2} \right)} + 21 y \cos{\left(x^{3} \right)}} " /> </p> </p>