Question

find the derivative the following ways. a. using the product rule. b. by expanding the product first. verify that your answer agrees with part (a). h(z)=(z³ - 2z² + 4z)(z - 5). a. use the product rule to find the derivative of the given function. select the correct choice and fill in the answer box(es) to complete your choice. a. the derivative is (z - 5)( )+( ) b. the derivative is (z³ - 2z² + 4z)( ) c. the derivative is ( )(z - 5)+(z³ - 2z² + 4z)( ) d. the derivative is (z - 5)(z³ - 2z² + 4z)+( ) e. the derivative is (z - 5)(z³ - 2z² + 4z)( )

Explanation:

Step1: Recall the Product Rule

The Product Rule states that if \(y = f(z)g(z)\), then \(y'=f'(z)g(z)+f(z)g'(z)\). Let \(f(z)=z - 5\) and \(g(z)=z^{3}-2z^{2}+4z\).

Step2: Find the derivatives of \(f(z)\) and \(g(z)\)

The derivative of \(f(z)=z - 5\) is \(f'(z)=1\). The derivative of \(g(z)=z^{3}-2z^{2}+4z\) using the power - rule \((x^n)'=nx^{n - 1}\) is \(g'(z)=3z^{2}-4z + 4\).

Step3: Apply the Product Rule

\[

$$\begin{align*} h'(z)&=f'(z)g(z)+f(z)g'(z)\\ &=1\times(z^{3}-2z^{2}+4z)+(z - 5)(3z^{2}-4z + 4) \end{align*}$$

\]

Answer:

C. The derivative is \((1)(z^{3}-2z^{2}+4z)+(z - 5)(3z^{2}-4z + 4)\)