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

From Math Wiki
Jump to navigation Jump to search
 
Line 33: Line 33:
 
!Step 1:  
 
!Step 1:  
 
|-
 
|-
|We need to start by identifying the two functions that are being multiplied together so we can apply the product rule.
+
|We need to start by identifying the two functions that are being multiplied together so we can apply the product rule. Let's call <math style="vertical-align: -20%">g(x)\,=\,2x^3,\,</math> and <math style="vertical-align: -20%">\,h(x) \, = \, e^{3x + 5}</math>, so <math style="vertical-align: -20%">f(x)=g(x)\cdot h(x)</math>.
|-
 
|
 
::<math>g(x)\,=\,2x^3,\,</math> and <math>\,h(x) \, = \, e^{3x + 5}</math>
 
 
|}
 
|}
  
Line 42: Line 39:
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
|We can now apply the three advanced techniques.This allows us to see that  
+
|We can now apply the advanced techniques.This allows us to see that  
 
|-
 
|-
 
|
 
|
<math>\begin{array}{rcl}
+
::<math>\begin{array}{rcl}
 
f'(x)&=&2(x^3)' e^{3x+5}+2x^3(e^{3x+5})' \\
 
f'(x)&=&2(x^3)' e^{3x+5}+2x^3(e^{3x+5})' \\
 
&=&6x^2e^{3x+5}+2x^3(3e^{3x+5})\\
 
&=&6x^2e^{3x+5}+2x^3(3e^{3x+5})\\
& = &6x^2e^{3x+5}+6x^3e^{3x+5}
+
& = &6x^2e^{3x+5}+6x^3e^{3x+5}.
 
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
Line 56: Line 53:
 
|-
 
|-
 
|
 
|
<math>6x^2e^{3x+5}+6x^3e^{3x+5}
+
::<math>f'(x)\,=\,6x^2e^{3x+5}+6x^3e^{3x+5}.
 
</math>
 
</math>
 
|}
 
|}
  
 
[[022_Exam_2_Sample_B|'''<u>Return to Sample Exam</u>''']]
 
[[022_Exam_2_Sample_B|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 07:50, 17 May 2015

Find the derivative of .


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

    

The Product Rule: If and are differentiable functions, then

    
Additionally, we will need our power rule for differentiation:
for ,
as well as the derivative of the exponential function, :

 Solution:

Step 1:  
We need to start by identifying the two functions that are being multiplied together so we can apply the product rule. Let's call and , so .
Step 2:  
We can now apply the advanced techniques.This allows us to see that
Final Answer:  

Return to Sample Exam