Questions: Use the product rule to find the derivative of the given function. Find the derivative by expanding the product first. h(z)=(7-z^2)(z^3-4 z+2) a. Use the product rule to find the derivative of the given function. Select the correct answer below and fill in the answer box(es) to complete your choice A. The derivative is (z^3-4 z+2)(). B. The derivative is (7-z^2)(z^3-4 z+2)(). C. The derivative is (7-z^2)(()+((z^3-4 z+2)())). D. The derivative is (7-z^2)(z^3-4 z+2)+(). E. The derivative is (7-z^2)().

Use the product rule to find the derivative of the given function.
Find the derivative by expanding the product first.

h(z)=(7-z^2)(z^3-4 z+2)

a. Use the product rule to find the derivative of the given function. Select the correct answer below and fill in the answer box(es) to complete your choice
A. The derivative is (z^3-4 z+2)().
B. The derivative is (7-z^2)(z^3-4 z+2)().
C. The derivative is (7-z^2)(()+((z^3-4 z+2)())).
D. The derivative is (7-z^2)(z^3-4 z+2)+().
E. The derivative is (7-z^2)().

Transcript text: Use the product rule to find the derivative of the given function. Find the derivative by expanding the product first. \[ h(z)=\left(7-z^{2}\right)\left(z^{3}-4 z+2\right) \] a. Use the product rule to find the derivative of the given function. Select the correct answer below and fill in the answer box(es) to complete your choice A. The derivative is $\left(z^{3}-4 z+2\right)(\square)$. B. The derivative is $\left(7-z^{2}\right)\left(z^{3}-4 z+2\right)(\square)$. C. The derivative is $\left(7-z^{2}\right)\left(\square+\left(z^{3}-4 z+2\right)(\square)\right.$. D. The derivative is $\left(7-z^{2}\right)\left(z^{3}-4 z+2\right)+(\square$. E. The derivative is $\left(7-z^{2}\right)(\square)$.

Solution

Solution Steps

Step 1: Using the Product Rule

To find the derivative of $h(z)$ using the product rule, we first identify $p(z) = a - z^2$ and $q(z) = z^3 - 4z + 2$. The derivatives are $p'(z) = -2z$ and $q'(z) = 3z^2 - 4$. Applying the product rule, $h'(z) = (-2z)(z^3 - 4z + 2) + (a - z^2)(3z^2 - 4)$, which simplifies to $h'(z) = (-2z)(z^3 - 4z + 2) + (a - z^2)(3z^2 - 4)$.

Step 2: By Expanding the Product First

First, we expand $h(z) = (a - z^2)(z^3 - bz + c)$ to get $h(z) = 7_z^3 - 28_z + 14 - z^5 + 4_z^3 - 2_z^2$. Simplifying, we get $h(z) = -z^5 + 11_z^3 - 2_z^2 - 28_z + 14$. Taking the derivative, $h'(z) = -5z^4 + 33_z^2 - 2_2_z - 28$, which simplifies to $h'(z) = -5z^4 + 33z^2 - 4z - 28$.