009A Sample Final 1, Problem 4 Detailed Solution

If $y=\cos ^{-1}(2x)$ compute ${\frac {dy}{dx}}$ and find the equation for the tangent line at $x_{0}={\frac {\sqrt {3}}{4}}.$

You may leave your answers in point-slope form.

Background Information:
1. Chain Rule
${\frac {d}{dx}}(f(g(x)))=f'(g(x))g'(x)$
2. Recall
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}(\cos^{-1}(x))=\frac{-1}{\sqrt{1-x^2}}}
3. The equation of the tangent line to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} at the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b)} is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=m(x-a)+b} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=f'(a).}

Solution:

Step 1:
First, we compute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dx}.}
Using the Chain Rule, we get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\frac{dy}{dx}} & = & \displaystyle{\frac{-1}{\sqrt{1-(2x)^2}}(2x)'}\\ &&\\ & = & \displaystyle{\frac{-2}{\sqrt{1-4x^2}}.} \end{array}}

Step 2:
To find the equation of the tangent line, we first find the slope of the line.
Using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0=\frac{\sqrt{3}}{4}} in the formula for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dx}} from Step 1, we get
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {m}&=&\displaystyle {\frac {-2}{\sqrt {1-4({\frac {\sqrt {3}}{4}})^{2}}}}\\&&\\&=&\displaystyle {\frac {-2}{\sqrt {\frac {1}{4}}}}\\&&\\&=&\displaystyle {-4.}\end{array}}}

Step 3:
To get a point on the line, we plug in Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x_{0}={\frac {\sqrt {3}}{4}}} into the equation given.
So, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {y_{0}}&=&\displaystyle {\cos ^{-1}{\bigg (}2{\frac {\sqrt {3}}{4}}{\bigg )}}\\&&\\&=&\displaystyle {\cos ^{-1}{\bigg (}{\frac {\sqrt {3}}{2}}{\bigg )}}\\&&\\&=&\displaystyle {{\frac {\pi }{6}}.}\end{array}}}
Thus, the equation of the tangent line is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=-4{\bigg (}x-{\frac {\sqrt {3}}{4}}{\bigg )}+{\frac {\pi }{6}}.}

Final Answer:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {dy}{dx}}={\frac {-2}{\sqrt {1-4x^{2}}}}}
$y=-4{\bigg (}x-{\frac {\sqrt {3}}{4}}{\bigg )}+{\frac {\pi }{6}}$

Navigation menu