Use the product rule to find the derivative of the following function.𝑦=9𝑥(𝑥2−8𝑥+6)4

The product rule states that the derivative of a product of two functions is the derivative of the first times the second plus the first times the derivative of the second.

Let's denote: f(x) = 9x g(x) = (x^2 - 8x + 6)^4

We need to find f'(x) and g'(x).

f'(x) = derivative of 9x = 9

To find g'(x), we need to use the chain rule because g is a composition of two functions: h(x) = x^2 - 8x + 6 and j(x) = x^4. The chain rule states that the derivative of a composition of functions is the derivative of the outer function times the derivative of the inner function.

So, g'(x) = 4*(x^2 - 8x + 6)^3 * (2x - 8) = 4*(2x - 8)*(x^2 - 8x + 6)^3

Now, we can apply the product rule:

y' = f'(x)g(x) + f(x)g'(x) = 9*(x^2 - 8x + 6)^4 + 9x4(2x - 8)(x^2 - 8x + 6)^3 = 9(x^2 - 8x + 6)^4 + 36x*(2x - 8)*(x^2 - 8x + 6)^3

This is the derivative of the given function.

The product rule for differentiation 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.

Given the function y = 9x(x^2 - 8x + 6)^4, we can identify two functions:

1. f(x) = 9x
2. g(x) = (x^2 - 8x + 6)^4

We first find the derivatives of these two functions:

f'(x) = 9 g'(x) = 4(x^2 - 8x + 6)^3 * (2x - 8) (by the chain rule)

Then, we apply the product rule:

y' = f'(x)g(x) + f(x)g'(x) = 9 * (x^2 - 8x + 6)^4 + 9x * 4(x^2 - 8x + 6)^3 * (2x - 8)

This is the derivative of the given function.