Paul's Online Notes
Home / Calculus I / Derivatives / Product and Quotient Rule
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.
Assignment Problems Notice
Please do not email me to get solutions and/or answers to these problems. I will not give them out under any circumstances nor will I respond to any requests to do so. The intent of these problems is for instructors to use them for assignments and having solutions/answers easily available defeats that purpose.

### Section 3.4 : Product and Quotient Rule

For problems 1 – 7 use the Product Rule or the Quotient Rule to find the derivative of the given function.

1. $$h\left( z \right) = \left( {2 - \sqrt z } \right)\left( {3 + 8\,\,\sqrt[3]{{{z^2}}}} \right)$$
2. $$\displaystyle f\left( x \right) = \left( {x - \frac{2}{x}} \right)\left( {7 - 2{x^3}} \right)$$
3. $$y = \left( {{x^2} - 5x + 1} \right)\left( {12 + 2x - {x^3}} \right)$$
4. $$\displaystyle g\left( x \right) = \frac{{\sqrt[3]{x}}}{{1 + {x^2}}}$$
5. $$\displaystyle Z\left( y \right) = \frac{{4y - {y^2}}}{{6 - y}}$$
6. $$\displaystyle V\left( t \right) = \frac{{1 - 10t + {t^2}}}{{5t + 2{t^3}}}$$
7. $$\displaystyle f\left( w \right) = \frac{{\left( {1 - 4w} \right)\left( {2 + w} \right)}}{{3 + 9w}}$$

For problems 8 – 12 use the fact that $$f\left( { - 3} \right) = 12$$, $$f'\left( { - 3} \right) = 9$$, $$g\left( { - 3} \right) = - 4$$, $$g'\left( { - 3} \right) = 7$$, $$h\left( { - 3} \right) = - 2$$ and $$h'\left( { - 3} \right) = 5$$ determine the value of the indicated derivative.

1. $${\left( {f\,g} \right)^\prime }\left( { - 3} \right)$$
2. $${\left( {\frac{h}{g}} \right)^\prime }\left( { - 3} \right)$$
3. $$\displaystyle {\left( {\frac{{f\,g}}{h}} \right)^\prime }\left( { - 3} \right)$$
4. If $$y = \left[ {x - f\left( x \right)} \right]h\left( x \right)$$ determine $$\displaystyle {\left. {\frac{{dy}}{{dx}}} \right|_{x = - 3}}$$.
5. If $$\displaystyle y = \frac{{1 - g\left( x \right)h\left( x \right)}}{{x + f\left( x \right)}}$$ determine $$\displaystyle {\left. {\frac{{dy}}{{dx}}} \right|_{x = - 3}}$$.
6. Find the equation of the tangent line to $$f\left( x \right) = \left( {8 - {x^2}} \right)\left( {1 + x + {x^2}} \right)$$ at $$x = - 2$$.
7. Find the equation of the tangent line to $$\displaystyle f\left( x \right) = \frac{{4 - {x^3}}}{{x + 2{x^2}}}$$ at $$x = 1$$.
8. Determine where $$\displaystyle g\left( z \right) = \frac{{2 - z}}{{12 + {z^2}}}$$ is increasing and decreasing.
9. Determine where $$R\left( x \right) = \left( {3 - x} \right)\left( {1 - 2x + {x^2}} \right)$$ is increasing and decreasing.
10. Determine where $$\displaystyle h\left( t \right) = \frac{{7t - {t^2}}}{{1 + 2{t^2}}}$$ is increasing and decreasing.
11. Determine where $$\displaystyle f\left( x \right) = \frac{{1 + x}}{{1 - x}}$$ is increasing and decreasing.
12. Derive the formula for the Product Rule for four functions. ${\left( {f\,g\,h\,w} \right)^\prime } = f'\,g\,h\,w + f\,g'\,h\,w + f\,g\,h'\,w + f\,g\,h\,w'$