009A Sample Final 2, Problem 3

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}.}

(a)  $y={\bigg (}{\frac {x^{2}+3}{x^{2}-1}}{\bigg )}^{3}$

(b)  $y=x\cos({\sqrt {x+1}})$

(c)  $y=\sin ^{-1}x$

Foundations:
1. Product Rule
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}(f(x)g(x))=f(x)g'(x)+f'(x)g(x)}
2. Quotient Rule
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}\bigg(\frac{f(x)}{g(x)}\bigg)=\frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^2}}
3. Chain Rule
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}(f(g(x)))=f'(g(x))g'(x)}

Solution:

(a)

Step 1:
Using the Chain Rule, 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 \frac{dy}{dx}=3\bigg(\frac{x^2+3}{x^2-1}\bigg)^2\bigg(\frac{x^2+3}{x^2-1}\bigg)'.}

Step 2:
Now, using the Quotient Rule, 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{\frac{dy}{dx}} & = & \displaystyle{3\bigg(\frac{x^2+3}{x^2-1}\bigg)^2\bigg(\frac{x^2+3}{x^2-1}\bigg)'}\\ &&\\ & = & \displaystyle{3\bigg(\frac{x^2+3}{x^2-1}\bigg)^2\bigg(\frac{(x^2-1)(x^2+3)'-(x^2+3)(x^2-1)'}{(x^2-1)^2}\bigg)}\\ &&\\ & = & \displaystyle{3\bigg(\frac{x^2+3}{x^2-1}\bigg)^2\bigg(\frac{(x^2-1)(2x)-(x^2+3)(2x)}{(x^2-1)^2}\bigg)}\\ &&\\ & = & \displaystyle{3\bigg(\frac{x^2+3}{x^2-1}\bigg)^2\bigg(\frac{2x^3-2x-2x^3-6x}{(x^2-1)^2}\bigg)}\\ &&\\ & = & \displaystyle{\frac{3(x^2+3)^2(-8x)}{(x^2-1)^4}.} \end{array}}

(b)

Step 1:
Using the Product Rule, 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 \frac{dy}{dx}=x(\cos(\sqrt{x+1}))'+(x)'\cos(\sqrt{x+1}).}

Step 2:
Now, 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{x(\cos(\sqrt{x+1}))'+(x)'\cos(\sqrt{x+1})}\\ &&\\ & = & \displaystyle{x(-\sin(\sqrt{x+1}))(\sqrt{x+1})'+(1)\cos(\sqrt{x+1})}\\ &&\\ & = & \displaystyle{-x\sin(\sqrt{x+1})\frac{1}{2\sqrt{x+1}}(x+1)'+\cos(\sqrt{x+1})}\\ &&\\ & = & \displaystyle{\frac{-x\sin(\sqrt{x+1})}{2\sqrt{x+1}}+\cos(\sqrt{x+1}).} \end{array}}

(c)

Step 1:
Let  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)=\sin(x)}   and  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 g(x)=\sin^{-1}x.}
These functions are inverses of each other since
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(g(x))=x}   and   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 g(f(x))=x.}
Therefore,
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{g'(x)} & = & \displaystyle{\frac{1}{f'(g(x))}}\\ &&\\ & = & \displaystyle{\frac{1}{\cos(\sin^{-1} x)}.} \end{array}}
Now, let  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=\sin^{-1}(x).}   Then,  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=\sin(y).}
So,  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 \cos(\sin^{-1} x)=\cos(y).}
Therefore,
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 g'(x)=\frac{1}{\cos(y)}.}

Step 2:
Now, since
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 \cos^2 y+\sin^2 y =1,}
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{\cos(y)} & = & \displaystyle{\sqrt{1-\sin^2 y}}\\ &&\\ & = & \displaystyle{\sqrt{1-x^2}.} \end{array}}
Hence,
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 g'(x)=\frac{1}{\sqrt{1-x^2}}.}

Final Answer:
(a)    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{3(x^2+3)^2(-8x)}{(x^2-1)^4}}
(b)    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{-x\sin(\sqrt{x+1})}{2\sqrt{x+1}}+\cos(\sqrt{x+1})}
(c)    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{1}{\sqrt{1-x^2}}}

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