009A Sample Final 1, Problem 4

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.

Foundations:
1. Chain Rule
${\frac {d}{dx}}(f(g(x)))=f'(g(x))g'(x)$
2. Recall
${\frac {d}{dx}}(\cos ^{-1}(x))={\frac {-1}{\sqrt {1-x^{2}}}}$
3. The equation of the tangent line to  $f(x)$  at the point  $(a,b)$  is
$y=m(x-a)+b$  where  $m=f'(a).$

Solution:

Step 1:
First, we compute  ${\frac {dy}{dx}}.$
Using the Chain Rule, we get
${\begin{array}{rcl}\displaystyle {\frac {dy}{dx}}&=&\displaystyle {{\frac {-1}{\sqrt {1-(2x)^{2}}}}(2x)'}\\&&\\&=&\displaystyle {{\frac {-2}{\sqrt {1-4x^{2}}}}.}\end{array}}$

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 (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{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 (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}}   into the equation given.
So, we have
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{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 (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=-4\bigg(x-\frac{\sqrt{3}}{4}\bigg)+\frac{\pi}{6}.}

Final Answer:
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}=\frac{-2}{\sqrt{1-4x^2}}}
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=-4\bigg(x-\frac{\sqrt{3}}{4}\bigg)+\frac{\pi}{6}}

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