Difference between revisions of "009A Sample Midterm 2, Problem 5"
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|First, we use the Chain Rule to get | |First, we use the Chain Rule to get | ||
|- | |- | ||
| − | | <math>f'(x)=3\tan^2( | + | | <math>f'(x)=3\tan^2(7x^2+5)(\tan(7x^2+5))'.</math> |
|} | |} | ||
| Line 31: | Line 31: | ||
!Step 2: | !Step 2: | ||
|- | |- | ||
| − | | | + | |Now, we use the Chain Rule again to get |
|- | |- | ||
| | | | ||
| + | <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> | ||
|} | |} | ||
| Line 90: | Line 97: | ||
!Final Answer: | !Final Answer: | ||
|- | |- | ||
| − | |'''(a)''' | + | | '''(a)''' <math>3\tan^2(7x^2+5)\sec^2(7x^2+5)(14x)</math> |
|- | |- | ||
|'''(b)''' | |'''(b)''' | ||
Revision as of 12:35, 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)=\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)} }
| Foundations: |
|---|
| 1. Chain Rule |
| 2. Derivatives of trig/ln |
| 3. Quotient Rule |
Solution:
(a)
| Step 1: |
|---|
| First, we use the Chain 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 f'(x)=3\tan^2(7x^2+5)(\tan(7x^2+5))'.} |
| Step 2: |
|---|
| Now, we use the Chain Rule again 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{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}} |
(b)
| Step 1: |
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| Step 2: |
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(c)
| Step 1: |
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| Step 2: |
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| 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 3\tan^2(7x^2+5)\sec^2(7x^2+5)(14x)} |
| (b) |
| (c) |