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) b. expand the product (z³ - 2z²+4z)(z - 5)=z⁴ - 7z³+14z² - 20z (simplify your answer.) select the correct choice and fill in the answer box(es) to complete your choice. a. the derivatives are different. using the product rule, the derivative is, and when expanding the product first, the derivative is. b. using either approach, d/dz((z³ - 2z²+4z)(z - 5)) = c. the derivative is (3z² - 4z + 4)(z - 5)+(z³ - 2z²+4z)(1) d. the derivative is (z - 5)(z³ - 2z²+4z)+( ) e. the derivative is (z - 5)(3z² - 4z + 4)( ) 3.4.9 question | 1/1 pt 3.4.10 question | 0/1 pt 3.4.24 - setup & solve question | 0/1 pt 3.4.23 question | 0/1 pt 3.4.43 question | 0/1 pt

Explanation:

Step1: Recall the product - rule

The product - rule states that if \(h(z)=f(z)g(z)\), then \(h^\prime(z)=f^\prime(z)g(z)+f(z)g^\prime(z)\). Let \(f(z)=z^{3}-2z^{2}+4z\) and \(g(z)=z - 5\). First, find the derivatives of \(f(z)\) and \(g(z)\). The derivative of \(f(z)\) is \(f^\prime(z)=\frac{d}{dz}(z^{3}-2z^{2}+4z)=3z^{2}-4z + 4\), and the derivative of \(g(z)\) is \(g^\prime(z)=\frac{d}{dz}(z - 5)=1\).

Step2: Apply the product - rule

Using the product - rule \(h^\prime(z)=(3z^{2}-4z + 4)(z - 5)+(z^{3}-2z^{2}+4z)(1)\).
Expand \((3z^{2}-4z + 4)(z - 5)\):
\[

$$\begin{align*} (3z^{2}-4z + 4)(z - 5)&=3z^{2}\cdot z-3z^{2}\cdot5-4z\cdot z + 4z\cdot5+4z-20\\ &=3z^{3}-15z^{2}-4z^{2}+20z + 4z-20\\ &=3z^{3}-19z^{2}+24z-20 \end{align*}$$

\]
And \((z^{3}-2z^{2}+4z)(1)=z^{3}-2z^{2}+4z\).
Then \(h^\prime(z)=3z^{3}-19z^{2}+24z-20+z^{3}-2z^{2}+4z=4z^{3}-21z^{2}+28z - 20\).

Step3: Expand the product first

Expand \(h(z)=(z^{3}-2z^{2}+4z)(z - 5)=z^{3}\cdot z-5z^{3}-2z^{2}\cdot z + 10z^{2}+4z\cdot z-20z=z^{4}-5z^{3}-2z^{3}+10z^{2}+4z^{2}-20z=z^{4}-7z^{3}+14z^{2}-20z\).
Then find the derivative of \(h(z)\): \(h^\prime(z)=\frac{d}{dz}(z^{4}-7z^{3}+14z^{2}-20z)=4z^{3}-21z^{2}+28z - 20\).

Answer:

\(4z^{3}-21z^{2}+28z - 20\)