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

Revision as of 16:35, 15 May 2015

Find the derivative of $f(x)\,=\,2x^{3}e^{3x+5}$ .

Foundations:
This problem requires several advanced rules of differentiation. In particular, you need
The Chain Rule: If $f$ and $g$ are differentiable functions, then
$(f\circ g)'(x)=f'(g(x))\cdot g'(x).$
The Product Rule: If $f$ and $g$ are differentiable functions, then
$(fg)'(x)=f'(x)\cdot g(x)+f(x)\cdot g'(x).$
Additionally, we will need our power rule for differentiation:
$\left(x^{n}\right)'\,=\,nx^{n-1},$ for $n\neq 0$ ,
as well as the derivative of the exponential function, $e^{x}$ :
$\left(e^{f(x)}\right)'\,=\,f'(x)\cdot e^{f(x)}.$

Solution:

Step 1:
We need to start by identifying the two functions that are being multiplied together so we can apply the product rule.
$g(x)\,=\,2x^{3},$
and $h(x)\,=\,e^{3x+5}$

Step 1:
We can now apply the three advanced techniques.This allows us to see that
${\begin{array}{rcl}f'(x)&=&2(x^{3})'e^{3x+5}+2x^{3}(e^{3x+5})'\\&=&6x^{2}e^{3x+5}+6x^{3}e^{3x+5}\end{array}}$

6x^{2}e^{3x+5}+6x^{3}e^{3x+5}

Final Answer:

@@ Line 24: / Line 24: @@
 |-
 |
-::<math>\left(e^{f(x)}\right)'\,=\,\left(f(x)\right)'\cdot e^{f(x)}.</math>
+::<math>\left(e^{f(x)}\right)'\,=\,f'(x)\cdot e^{f(x)}.</math>
 |<br>
 |}
@@ Line 33: / Line 33: @@
 !Step 1: &nbsp;
 |-
-|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
+|We need to start by identifying the two functions that are being multiplied together so we can apply the product rule.
 |-
 |
-::<math>f(x)\,=\,\frac{(x+5)(x-1)}{x},</math>
+::<math>g(x)\,=\,2x^3,</math>
 |-
-|we then have&thinsp; <math style="vertical-align: -21%">y\,=\,g\circ f(x)\,=\,g\left(f(x)\right).</math>
+|and <math>h(x) \, = \, e^{3x + 5}</math>
+|}
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+!Step 1: &nbsp;
+|-
+|We can now apply the three advanced techniques.This allows us to see that
+|-
+|
+<math>\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}</math>
+|}
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+!Final Answer: &nbsp;
+|-
+<math>6x^2e^{3x+5}+6x^3e^{3x+5}
+</math>
 |}