Difference between revisions of "022 Exam 2 Sample B, Problem 3"

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::<math>\left(e^{f(x)}\right)'\,=\,\left(f(x)\right)'\cdot e^{f(x)}.</math>
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::<math>\left(e^{f(x)}\right)'\,=\,f'(x)\cdot e^{f(x)}.</math>
 
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!Step 1: &nbsp;
 
!Step 1: &nbsp;
 
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|We need to identify the composed functions in order to apply the chain rule. Note that if we set <math style="vertical-align: -21%">g(x)\,=\,\ln x</math>, and
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|We need to start by identifying the two functions that are being multiplied together so we can apply the product rule.
 
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::<math>f(x)\,=\,\frac{(x+5)(x-1)}{x},</math>
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::<math>g(x)\,=\,2x^3,</math>
 
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|we then have&thinsp; <math style="vertical-align: -21%">y\,=\,g\circ f(x)\,=\,g\left(f(x)\right).</math>
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|and <math>h(x) \, = \, e^{3x + 5}</math>
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 1: &nbsp;
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|We can now apply the three advanced techniques.This allows us to see that
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<math>\begin{array}{rcl}
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f'(x)&=&2(x^3)' e^{3x+5}+2x^3(e^{3x+5})' \\
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&=&6x^2e^{3x+5}+6x^3e^{3x+5}
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\end{array}</math>
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Final Answer: &nbsp;
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<math>6x^2e^{3x+5}+6x^3e^{3x+5}
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</math>
 
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Revision as of 16:35, 15 May 2015

Find the derivative of .


Foundations:  
This problem requires several advanced rules of differentiation. In particular, you need
The Chain Rule: If 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} and 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} are differentiable functions, then

     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\circ g)'(x) = f'(g(x))\cdot g'(x).}

The Product Rule: If 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} and 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} are differentiable functions, then

     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 (fg)'(x) = f'(x)\cdot g(x)+f(x)\cdot g'(x).}
Additionally, we will need our power rule for differentiation:
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 \left(x^n\right)'\,=\,nx^{n-1},} 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 n\neq 0} ,
as well as the derivative of the exponential function, 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 e^x} :
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 \left(e^{f(x)}\right)'\,=\,f'(x)\cdot e^{f(x)}.}

 Solution:

Step 1:  
We need to start by identifying the two functions that are being multiplied together so we can apply the product rule.
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)\,=\,2x^3,}
and 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) \, = \, e^{3x + 5}}
Step 1:  
We can now apply the three advanced techniques.This allows us to see that

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} f'(x)&=&2(x^3)' e^{3x+5}+2x^3(e^{3x+5})' \\ &=&6x^2e^{3x+5}+6x^3e^{3x+5} \end{array}}

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 6x^2e^{3x+5}+6x^3e^{3x+5} }
Final Answer:  
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