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.

Section 3-4 : Product and Quotient Rule

For problems 1 – 6 use the Product Rule or the Quotient Rule to find the derivative of the given function.

1. $$f\left( t \right) = \left( {4{t^2} - t} \right)\left( {{t^3} - 8{t^2} + 12} \right)$$ Solution
2. $$y = \left( {1 + \sqrt {{x^3}} } \right)\,\left( {{x^{ - 3}} - 2\sqrt[3]{x}} \right)$$ Solution
3. $$h\left( z \right) = \left( {1 + 2z + 3{z^2}} \right)\left( {5z + 8{z^2} - {z^3}} \right)$$ Solution
4. $$\displaystyle g\left( x \right) = \frac{{6{x^2}}}{{2 - x}}$$ Solution
5. $$\displaystyle R\left( w \right) = \frac{{3w + {w^4}}}{{2{w^2} + 1}}$$ Solution
6. $$\displaystyle f\left( x \right) = \frac{{\sqrt x + 2x}}{{7x - 4{x^2}}}$$ Solution
7. If$$f\left( 2 \right) = - 8$$, $$f'\left( 2 \right) = 3$$, $$g\left( 2 \right) = 17$$ and $$g'\left( 2 \right) = - 4$$ determine the value of $${\left( {f\,g} \right)^\prime }\left( 2 \right)$$. Solution
8. If $$f\left( x \right) = {x^3}g\left( x \right)$$, $$g\left( { - 7} \right) = 2$$, $$g'\left( { - 7} \right) = - 9$$ determine the value of $$f'\left( { - 7} \right)$$. Solution
9. Find the equation of the tangent line to $$f\left( x \right) = \left( {1 + 12\sqrt x } \right)\left( {4 - {x^2}} \right)$$ at $$x = 9$$. Solution
10. Determine where $$\displaystyle f\left( x \right) = \frac{{x - {x^2}}}{{1 + 8{x^2}}}$$ is increasing and decreasing. Solution
11. Determine where $$V\left( t \right) = \left( {4 - {t^2}} \right)\left( {1 + 5{t^2}} \right)$$ is increasing and decreasing. Solution