009A Sample Midterm 2, Problem 5 Detailed Solution

Find the derivatives of the following functions. Do not simplify.

(a)   Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=\tan ^{3}(7x^{2}+5)}

(b)   Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g(x)=\sin(\cos(e^{x}))}

(c)   Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle h(x)={\frac {(5x^{2}+7x)^{3}}{\ln(x^{2}+1)}}}

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Background Information:
1. Chain Rule
${\frac {d}{dx}}(f(g(x)))=f'(g(x))g'(x)$
2. Trig Derivatives
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {d}{dx}}(\sin x)=\cos x,\quad {\frac {d}{dx}}(\cos x)=-\sin x}
3. Quotient Rule
${\frac {d}{dx}}{\bigg (}{\frac {f(x)}{g(x)}}{\bigg )}={\frac {g(x)f'(x)-f(x)g'(x)}{(g(x))^{2}}}$
4. Derivative of natural logarithm
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}(\ln x)=\frac{1}{x}}

Solution:

(a)

Step 1:
First, we use the Chain Rule to 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 f'(x)=3\tan^2(7x^2+5)(\tan(7x^2+5))'.}

Step 2:
Now, we use the Chain Rule again to 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{f'(x)} & = & \displaystyle{3\tan^2(7x^2+5)(\tan(7x^2+5))'}\\ &&\\ & = & \displaystyle{3\tan^2(7x^2+5)\sec^2(7x^2+5)(7x^2+5)'}\\ &&\\ & = & \displaystyle{3\tan^2(7x^2+5)\sec^2(7x^2+5)(14x).} \end{array}}

(b)

Step 1:
First, we use the Chain Rule to 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 g'(x)=\cos(\cos(e^x))(\cos(e^x))'.}

Step 2:
Now, we use the Chain Rule again to 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{g'(x)} & = & \displaystyle{\cos(\cos(e^x))(\cos(e^x))'}\\ &&\\ & = & \displaystyle{\cos(\cos(e^x))(-\sin(e^x))(e^x)'}\\ &&\\ & = & \displaystyle{\cos(\cos(e^x))(-\sin(e^x))(e^x).} \end{array}}

(c)

Step 1:
First, we use the Quotient Rule to 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 h'(x)=\frac{\ln(x^2+1)((5x^2+7x)^3)'-(5x^2+7x)^3(\ln(x^2+1))'}{(\ln(x^2+1))^2}.}

Step 2:
Now, we use the Chain Rule to 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{h'(x)} & = & \displaystyle{\frac{\ln(x^2+1)((5x^2+7x)^3)'-(5x^2+7x)^3(\ln(x^2+1))'}{(\ln(x^2+1))^2}}\\ &&\\ & = & \displaystyle{\frac{\ln(x^2+1)3(5x^2+7x)^2(5x^2+7x)'-(5x^2+7x)^3\frac{1}{x^2+1}(x^2+1)'}{(\ln(x^2+1))^2}}\\ &&\\ & = & \displaystyle{\frac{\ln(x^2+1)3(5x^2+7x)^2(10x+7)-(5x^2+7x)^3\frac{1}{x^2+1}(2x)}{(\ln(x^2+1))^2}.} \end{array}}

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 f'(x)=3\tan^2(7x^2+5)\sec^2(7x^2+5)(14x)}
(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 g'(x)=\cos(\cos(e^x))(-\sin(e^x))(e^x)}
(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 h'(x)=\frac{\ln(x^2+1)3(5x^2+7x)^2(10x+7)-(5x^2+7x)^3\frac{1}{x^2+1}(2x)}{(\ln(x^2+1))^2}}

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