Difference between revisions of "009A Sample Midterm 1, Problem 4"

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!Step 1:    
 
!Step 1:    
 
|-
 
|-
−
|
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|Using the Product Rule, we have
 
|-
 
|-
−
|
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>f'(x)=(\sqrt{x})'(x^2+2)+\sqrt{x}(x^2+2)'.</math>
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
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|Now, we have
 
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 +
\displaystyle{f'(x)} & = & \displaystyle{(\sqrt{x})'(x^2+2)+\sqrt{x}(x^2+2)'}\\
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&&\\
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& = & \displaystyle{\bigg(\frac{1}{2}x^{-\frac{1}{2}}\bigg)(x^2+2)+\sqrt{x}(2x).}
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\end{array}</math>
 
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
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|'''(a)'''  
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|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; <math>\bigg(\frac{1}{2}x^{-\frac{1}{2}}\bigg)(x^2+2)+\sqrt{x}(2x)</math>
 
|-
 
|-
 
|'''(b)'''  
 
|'''(b)'''  

Revision as of 10:22, 16 February 2017

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

a)
b) where 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}
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{e^{-5x^3}}{\sqrt{x^2+1}}}


Foundations:  
1. Product Rule
2. Quotient Rule
3. Chain Rule

Solution:

(a)

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

(b)

Step 1:  
Step 2:  

(c)

Step 1:  
Step 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 \bigg(\frac{1}{2}x^{-\frac{1}{2}}\bigg)(x^2+2)+\sqrt{x}(2x)}
(b)
(c)

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