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### 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. ${\left( {f\,g\,h\,w} \right)^\prime } = f'\,g\,h\,w + f\,g'\,h\,w + f\,g\,h'\,w + f\,g\,h\,w'$