Difference between revisions of "009A Sample Midterm 3, Problem 5"
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!Step 2: | !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> | ||
|} | |} | ||
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!Final Answer: | !Final Answer: | ||
|- | |- | ||
| − | |'''(a)''' | + | | '''(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)''' | |'''(b)''' | ||
|} | |} | ||
[[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] | [[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 15:37, 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)=\frac{(3x-5)(-x^{-2}+4x)}{x^{\frac{4}{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)=\sqrt{x}+\frac{1}{\sqrt{x}}+\sqrt{\pi}} for 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. Quotient Rule |
| 2. Product Rule |
| 3. Power Rule |
Solution:
(a)
| Step 1: |
|---|
| Using the Quotient 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)=\frac{x^{\frac{4}{5}}((3x-5)(-x^{-2}+4x))'-(3x-5)(-x^{-2}+4x)(x^{\frac{4}{5}})'}{(x^{\frac{4}{5}})^2}.} |
| Step 2: |
|---|
| Now, we use the Product 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{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}} |
(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 \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}} |
| (b) |