Answer:
The derivative of the function h(z) = (6 - z^2)(z^3 - 5z + 2) is
h'(z) = -5z^4 + 23z^2 - 4z - 30.
Explanation:
To find the derivative of the function h(z) = (6 - z^2)(z^3 - 5z + 2), we can use the product rule.
The product rule states that if we have two functions u(z) and v(z), their product's derivative is given by:
(h(z))' = u'(z)v(z) + u(z)v'(z)
Let's find the derivative using the product rule:
First, we differentiate the first function u(z) = 6 - z^2:
u'(z) = -2z
Next, we differentiate the second function v(z) = z^3 - 5z + 2:
v'(z) = 3z^2 - 5
Now, we can apply the product rule:
(h(z))' = (6 - z^2)(3z^2 - 5) + (-2z)(z^3 - 5z + 2)
Simplifying the expression:
(h(z))' = (18z^2 - 30 - 3z^4 + 5z^2) + (-2z^4 + 10z^2 - 4z)
Combining like terms:
(h(z))' = -3z^4 + 18z^2 - 2z^4 + 5z^2 - 4z - 30
Simplifying further:
(h(z))' = -5z^4 + 23z^2 - 4z - 30
Therefore,
The derivative of the function h(z) = (6 - z^2)(z^3 - 5z + 2) is
h'(z) = -5z^4 + 23z^2 - 4z - 30.