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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
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. \[{\left( {f\,g\,h\,w} \right)^\prime } = f'\,g\,h\,w + f\,g'\,h\,w + f\,g\,h'\,w + f\,g\,h\,w'\]