009A Sample Midterm 3, Problem 5 - Revision history

MathAdmin at 22:25, 9 November 2017

2017-11-09T22:25:55Z

MathAdmin at 16:14, 7 November 2017

2017-11-07T16:14:15Z

MathAdmin at 02:49, 14 April 2017

2017-04-14T02:49:14Z

MathAdmin: Created page with " Find the derivatives of the following functions. Do not simplify. (a) $f(x)=\frac{(3x-5)(-x^{-..."$

2017-04-09T18:56:57Z

Created page with " Find the derivatives of the following functions. Do not simplify. (a) <math style="vertical-align: -16px">f(x)=\frac{(3x-5)(-x^{-..."

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

(a)  <math style="vertical-align: -16px">f(x)=\frac{(3x-5)(-x^{-2}+4x)}{x^{\frac{4}{5}}}</math>

(b)  <math>g(x)=\sqrt{x}+\frac{1}{\sqrt{x}}+\sqrt{\pi}</math>  for  <math style="vertical-align: 0px">x>0.</math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|'''1.''' '''Product Rule'''
|-
|        <math>\frac{d}{dx}(f(x)g(x))=f(x)g'(x)+f'(x)g(x)</math>
|-
|'''2.''' '''Quotient Rule'''
|-
|        <math>\frac{d}{dx}\bigg(\frac{f(x)}{g(x)}\bigg)=\frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^2}</math>
|-
|'''3.''' '''Power Rule'''
|-
|        <math>\frac{d}{dx}(x^n)=nx^{n-1}</math>
|}

'''Solution:'''

'''(a)'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|Using the Quotient Rule, we have
|-
|        <math>f'(x)=\frac{x^{\frac{4}{5}}((3x-5)(-x^{-2}+4x))'-(3x-5)(-x^{-2}+4x)(x^{\frac{4}{5}})'}{(x^{\frac{4}{5}})^2}.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|Now, we use the Product Rule to get
|-
|
        <math>\begin{array}{rcl}
\displaystyle{f'(x)} & = & \displaystyle{\frac{x^{\frac{4}{5}}((3x-5)(-x^{-2}+4x))'-(3x-5)(-x^{-2}+4x)(x^{\frac{4}{5}})'}{(x^{\frac{4}{5}})^2}}\\
&&\\
& = & \displaystyle{\frac{x^{\frac{4}{5}}[(3x-5)(-x^{-2}+4x)'+(3x-5)'(-x^{-2}+4x)]-(3x-5)(-x^{-2}+4x)(\frac{4}{5}x^{-\frac{1}{5}})}{(x^{\frac{4}{5}})^2}}\\
&&\\
& = & \displaystyle{\frac{x^{\frac{4}{5}}[(3x-5)(2x^{-3}+4)+(3)(-x^{-2}+4x)]-(3x-5)(-x^{-2}+4x)(\frac{4}{5}x^{-\frac{1}{5}})}{(x^{\frac{4}{5}})^2}.}
\end{array}</math>
|}

'''(b)'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|First, we have
|-
|        <math>g'(x)=(\sqrt{x})'+\bigg(\frac{1}{\sqrt{x}}\bigg)'+(\sqrt{\pi})'.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|Since  <math style="vertical-align: 0px">\pi</math>  is a constant,  <math style="vertical-align: -3px">\sqrt{\pi}</math>  is also a constant.
|-
|Hence,
|-
|        <math>(\sqrt{\pi})'=0.</math>
|-
|Therefore, we have
|-
|        <math>\begin{array}{rcl}
\displaystyle{g'(x)} & = & \displaystyle{(\sqrt{x})'+\bigg(\frac{1}{\sqrt{x}}\bigg)'+(\sqrt{\pi})'}\\
&&\\
& = & \displaystyle{\frac{1}{2}x^{-\frac{1}{2}}+-\frac{1}{2}x^{-\frac{3}{2}}+0}\\
&&\\
& = & \displaystyle{\frac{1}{2}x^{-\frac{1}{2}}+-\frac{1}{2}x^{-\frac{3}{2}}.}
\end{array}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|    '''(a)'''     <math>\frac{x^{\frac{4}{5}}[(3x-5)(2x^{-3}+4)+(3)(-x^{-2}+4x)]-(3x-5)(-x^{-2}+4x)(\frac{4}{5}x^{-\frac{1}{5}})}{(x^{\frac{4}{5}})^2}</math>
|-
|    '''(b)'''     <math>\frac{1}{2}x^{-\frac{1}{2}}+-\frac{1}{2}x^{-\frac{3}{2}}</math>
|}
[[009A_Sample_Midterm_3|'''Return to Sample Exam''']]

@@ Line 8: / Line 8: @@
 <hr>
-[[File:9ASM3P5.jpg|600px|thumb|center]]
+[[009A Sample Midterm 3, Problem 5 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 1: / Line 1: @@
 <span class="exam"> Find the derivatives of the following functions. Do not simplify.
-<span class="exam">(a)&nbsp; <math style="vertical-align: -16px">f(x)=\frac{(3x-5)(-x^{-2}+4x)}{x^{\frac{4}{5}}}</math>
+<span class="exam">(a)&nbsp; <math style="vertical-align: -16px">f(x)=\sin\bigg(\frac{x^{-3}}{e^{-x}}\bigg)</math>
-<span class="exam">(b)&nbsp; <math>g(x)=\sqrt{x}+\frac{1}{\sqrt{x}}+\sqrt{\pi}</math>&nbsp; for &nbsp;<math style="vertical-align: 0px">x>0.</math>
+<span class="exam">(b)&nbsp; <math style="vertical-align: -18px">g(x)=\sqrt{\frac{x^2+2}{x^2+4}}</math>
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+<span class="exam">(c)&nbsp; <math style="vertical-align: -6px">h(x)=(x+\cos^2x)^8</math>
-'''Solution:'''
+[[009A Sample Midterm 3, Problem 5 Detailed Solution|'''<u>Detailed Solution with Background Information</u>''']]
-'''(a)'''
+[[File:9ASM3P5.jpg|600px|thumb|center]]
 [[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 81: / Line 81: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; <math>\frac{x^{\frac{4}{5}}[(3x-5)(2x^{-3}+4)+(3)(-x^{-2}+4x)]-(3x-5)(-x^{-2}+4x)(\frac{4}{5}x^{-\frac{1}{5}})}{(x^{\frac{4}{5}})^2}</math>
+|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; <math>f'(x)=\frac{x^{\frac{4}{5}}[(3x-5)(2x^{-3}+4)+(3)(-x^{-2}+4x)]-(3x-5)(-x^{-2}+4x)(\frac{4}{5}x^{-\frac{1}{5}})}{(x^{\frac{4}{5}})^2}</math>
 |-
-|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; <math>\frac{1}{2}x^{-\frac{1}{2}}+-\frac{1}{2}x^{-\frac{3}{2}}</math>
+|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; <math>g'(x)=\frac{1}{2}x^{-\frac{1}{2}}+-\frac{1}{2}x^{-\frac{3}{2}}</math>
 |}
 [[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

009A Sample Midterm 3, Problem 5 - Revision history

MathAdmin at 22:25, 9 November 2017

MathAdmin at 16:14, 7 November 2017

MathAdmin at 02:49, 14 April 2017

MathAdmin: Created page with " Find the derivatives of the following functions. Do not simplify. (a) f(x)=\frac{(3x-5)(-x^{-..."

MathAdmin: Created page with " Find the derivatives of the following functions. Do not simplify. (a) $f(x)=\frac{(3x-5)(-x^{-..."$