Question

Ind the first and second derivatives of the function.

h(x) = (x² + 9)²(x - 9)
h'(x) =
h''(x) =

Answer 1

To find the first and second derivatives of the function h(x) = (x² + 9)²(x - 9), apply the product and chain rules. The first derivative is h'(x) = 2(x² + 9)(2x)(x - 9) + (x² + 9)². Then, differentiate h'(x) to obtain the second derivative, h''(x).

Step-by-step explanation:

The student has asked for help in finding the first and second derivatives of the function h(x) = (x² + 9)²(x - 9). To find the first derivative, h'(x), we will need to apply the product rule, as the function is a product of two functions. The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Applying the product rule:

1. Let u = (x² + 9)² and v = (x - 9)
2. Find u' (the derivative of u with respect to x) and v' (the derivative of v with respect to x)
3. Then, use the formula h'(x) = u'v + uv' to find the first derivative
4. To find the second derivative, h''(x), simply take the derivative of h'(x)

For the first derivative, u' will be 2(x² + 9)(2x) based on the chain rule, and v' will be 1. Plugging these into the product rule formula gives:

h'(x) = 2(x² + 9)(2x)(x - 9) + (x² + 9)²

For the second derivative, we take the derivative of h'(x) using the product and chain rules again