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Section 3.4 : Product and Quotient Rule

3. Use the Product Rule to find the derivative of \(h\left( z \right) = \left( {1 + 2z + 3{z^2}} \right)\left( {5z + 8{z^2} - {z^3}} \right)\) .

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There isn’t much to do here other than take the derivative using the product rule.

\[\begin{align*}h'\left( z \right) & = \left( {2 + 6z} \right)\left( {5z + 8{z^2} - {z^3}} \right) + \left( {1 + 2z + 3{z^2}} \right)\left( {5 + 16z - 3{z^2}} \right)\\ & = 5 + 36z + 90{z^2} + 88{z^3} - 15{z^4}\end{align*}\]

Note that we multiplied everything out to get a “simpler” answer.