Difference between revisions of "022 Exam 1 Sample A, Problem 8"

Latest revision as of 15:34, 2 April 2015

8. Find the derivative of the function $f(x)={\frac {(3x-1)^{2}}{x^{3}-7}}$ . You do not need to simplify your answer.

Foundations:
This problem involves some more advanced rules of differentiation. In particular, it requires
The Chain Rule: If $f$ and $g$ are differentiable functions, then
$(f\circ g)'(x)=f'(g(x))\cdot g'(x).$
The Quotient Rule: If $f$ and $g$ are differentiable functions and $g(x)\neq 0$ , then
$\left({\frac {f}{g}}\right)'(x)={\frac {f'(x)\cdot g(x)-f(x)\cdot g'(x)}{\left(g(x)\right)^{2}}}.$

Solution:

Note that we need to use chain rule to find the derivative of

\left(3x-1\right)^{2}

. Then we find

$f'(x)$	$=\,\,{\frac {\left[\left(3x-1\right)^{2}\right]'\cdot (x^{3}-7)\,\,-\,\,\left(3x-1\right)^{2}\cdot (x^{3}-7)'}{(x^{3}-7)^{2}}}$
	$=\,\,{\frac {\left[2\left(3x-1\right)\cdot 3\right]\cdot (x^{3}-7)\,\,-\,\,\left(3x-1\right)^{2}\cdot 3x^{2}}{(x^{3}-7)^{2}}}.$

@@ Line 1: / Line 1: @@
-<span style="font-size:135%">8. Find the derivative of the function <math style="vertical-align: -43%">f(x)=\frac{(3x-1)^{2}}{x^{3}-7}</math>.
+<span style="font-size:135%"><font face=Times Roman>8. Find the derivative of the function <math style="vertical-align: -43%">f(x)=\frac{(3x-1)^{2}}{x^{3}-7}</math>.
-You do not need to simplify your answer.
+You do not need to simplify your answer.</font face=Times Roman>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 19: / Line 19: @@
 |}
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+!Solution: &nbsp;
+|-
+|Note that we need to use chain rule to find the derivative of <math style="vertical-align: -25%;">\left(3x-1\right)^2</math>.  Then we find
+|-
+|<table>
+  <tr style="vertical-align: middle">
+    <td> &nbsp;&nbsp;&nbsp;&nbsp; <math style="vertical-align: -70%">f'(x)</math>&nbsp;&nbsp; </td>
+    <td><math>=\,\,\frac{\left[\left(3x-1\right)^{2}\right]'\cdot(x^{3}-7) \,\,-\,\, \left(3x-1\right)^{2}\cdot(x^{3}-7)'}{(x^{3}-7)^{2}}</math></td>
+  </tr>
+  <tr>
+    <td></td>
+    <td><math>=\,\,\frac{\left[2\left(3x-1\right)\cdot3\right]\cdot(x^{3}-7) \,\,-\,\, \left(3x-1\right)^{2}\cdot3x^{2}}{(x^{3}-7)^{2}}.</math></td>
+</table>
+|}
-&nbsp;'''Solution:'''
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+[[022_Exam_1_Sample_A|'''<u>Return to Sample Exam</u>''']]
-!Step 1: &nbsp;
-|-