Difference between revisions of "009A Sample Midterm 2, Problem 5"

Revision as of 09:12, 3 November 2017

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

(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)=\tan^3(7x^2+5) }

(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)=\sin(\cos(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{(5x^2+7x)^3}{\ln(x^2+1)} }

(insert picture of handwritten solution)

Detailed Solution for this Problem

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 <span class="exam">(c) &nbsp; <math style="vertical-align: -18px">h(x)=\frac{(5x^2+7x)^3}{\ln(x^2+1)} </math>
+<hr>
+(insert picture of handwritten solution)
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+[[009A Sample Midterm 2, Problem 5 Detailed Solution|'''<u>Detailed Solution for this Problem</u>''']]
-!Foundations: &nbsp;
-|-
-|'''1.''' '''Chain Rule'''
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{d}{dx}(f(g(x)))=f'(g(x))g'(x)</math>
-|-
-|'''2.''' '''Trig Derivatives'''
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{d}{dx}(\sin x)=\cos x,\quad\frac{d}{dx}(\cos x)=-\sin x</math>
-|-
-|'''3.''' '''Quotient Rule'''
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{d}{dx}\bigg(\frac{f(x)}{g(x)}\bigg)=\frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^2}</math>
-|-
-|'''4.''' '''Derivative of natural logarithm
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{d}{dx}(\ln x)=\frac{1}{x}</math>
-|}
-'''Solution:'''
-'''(a)'''
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
-|-
-|First, we use the Chain Rule to get
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>f'(x)=3\tan^2(7x^2+5)(\tan(7x^2+5))'.</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
-|-
-|Now, we use the Chain Rule again to get
-|-
-|
-&nbsp; &nbsp; &nbsp; &nbsp; <math>\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}</math>
-|}
-'''(b)'''
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
-|-
-|First, we use the Chain Rule to get
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>g'(x)=\cos(\cos(e^x))(\cos(e^x))'.</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
-|-
-|Now, we use the Chain Rule again to get
-|-
-|
-&nbsp; &nbsp; &nbsp; &nbsp; <math>\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}</math>
-|}
-'''(c)'''
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
-|-
-|First, we use the Quotient Rule to get
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>h'(x)=\frac{\ln(x^2+1)((5x^2+7x)^2)'-(5x^2+7x)^2(\ln(x^2+1))'}{(\ln(x^2+1))^2}.</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
-|-
-|Now, we use the Chain Rule to get
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
-\displaystyle{h'(x)} & = & \displaystyle{\frac{\ln(x^2+1)((5x^2+7x)^2)'-(5x^2+7x)^2(\ln(x^2+1))'}{(\ln(x^2+1))^2}}\\
-&&\\
-& = & \displaystyle{\frac{\ln(x^2+1)2(5x^2+7x)(5x^2+7x)'-(5x^2+7x)^2\frac{1}{x^2+1}(x^2+1)'}{(\ln(x^2+1))^2}}\\
-&&\\
-& = & \displaystyle{\frac{\ln(x^2+1)2(5x^2+7x)(10x+7)-(5x^2+7x)^2\frac{1}{x^2+1}(2x)}{(\ln(x^2+1))^2}.}
-\end{array}</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Final Answer: &nbsp;
-|-
-|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; <math>3\tan^2(7x^2+5)\sec^2(7x^2+5)(14x)</math>
-|-
-|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; <math>\cos(\cos(e^x))(-\sin(e^x))(e^x)</math>
-|-
-|&nbsp; &nbsp; '''(c)''' &nbsp; &nbsp; <math>\frac{\ln(x^2+1)2(5x^2+7x)(10x+7)-(5x^2+7x)^2\frac{1}{x^2+1}(2x)}{(\ln(x^2+1))^2}</math>
-|}
 [[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

Difference between revisions of "009A Sample Midterm 2, Problem 5"

Revision as of 09:12, 3 November 2017

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