Difference between revisions of "009A Sample Midterm 2, Problem 4"
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| Line 9: | Line 9: | ||
!Foundations: | !Foundations: | ||
|- | |- | ||
| − | | | + | |'''1.''' Product Rule |
| − | |||
| − | |||
| − | |||
|- | |- | ||
| − | | | + | |'''2.''' Quotient Rule |
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|} | |} | ||
| Line 25: | Line 21: | ||
!Step 1: | !Step 1: | ||
|- | |- | ||
| − | | | + | |Using the Product Rule, we have |
|- | |- | ||
| − | | | + | | <math>f'(x)=x^3(x^{\frac{4}{3}}-1)'+(x^3)'(x^{\frac{4}{3}}-1).</math> |
|} | |} | ||
| Line 33: | Line 29: | ||
!Step 2: | !Step 2: | ||
|- | |- | ||
| − | | | + | |Now, we have |
|- | |- | ||
| − | | | + | | <math>\begin{array}{rcl} |
| + | \displaystyle{f'(x)} & = & \displaystyle{x^3(x^{\frac{4}{3}}-1)'+(x^3)'(x^{\frac{4}{3}}-1)}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{x^3\bigg(\frac{4}{3}x^{\frac{1}{3}}\bigg)+(3x^2)(x^{\frac{4}{3}}-1).} | ||
| + | \end{array}</math> | ||
|} | |} | ||
| Line 67: | Line 67: | ||
!Final Answer: | !Final Answer: | ||
|- | |- | ||
| − | |'''(a)''' | + | | '''(a)''' <math>x^3\bigg(\frac{4}{3}x^{\frac{1}{3}}\bigg)+(3x^2)(x^{\frac{4}{3}}-1)</math> |
|- | |- | ||
|'''(b)''' | |'''(b)''' | ||
|} | |} | ||
[[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']] | [[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 12:19, 17 February 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)=x^3(x^{\frac{4}{3}}-1)}
- 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)=\frac{x^3+x^{-3}}{1+6x}} 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}
| Foundations: |
|---|
| 1. Product Rule |
| 2. Quotient 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)=x^3(x^{\frac{4}{3}}-1)'+(x^3)'(x^{\frac{4}{3}}-1).} |
| 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{x^3(x^{\frac{4}{3}}-1)'+(x^3)'(x^{\frac{4}{3}}-1)}\\ &&\\ & = & \displaystyle{x^3\bigg(\frac{4}{3}x^{\frac{1}{3}}\bigg)+(3x^2)(x^{\frac{4}{3}}-1).} \end{array}} |
(b)
| 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 x^3\bigg(\frac{4}{3}x^{\frac{1}{3}}\bigg)+(3x^2)(x^{\frac{4}{3}}-1)} |
| (b) |