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