Find the derivative.

{sqrt(x+8)}*(x^2+20x)

(x+8)^1/2(2x+20) minus (1/2)(x+8)^-1/2(x^2+20x)

I'm not feeling confident I am on the right track and I have a quiz tomorrow over this.

not bad, but why did you subtract?

d/dx (x+8)^(1/2) = (1/2)(x+8)^(-1/2)

your derivative is good, except for that.

To find the derivative of the given function, you can use the product rule, which states that if you have a function f(x) multiplied by another function g(x), the derivative of their product is f'(x)g(x) + f(x)g'(x).

Let's apply the product rule to the function sqrt(x+8) * (x^2 + 20x):

Step 1: Identify the functions f(x) and g(x).

In this case, f(x) = sqrt(x + 8) and g(x) = x^2 + 20x.

Step 2: Find the derivatives of f(x) and g(x).

The derivative of f(x) = sqrt(x + 8) can be found using the chain rule.

Since f(x) is in the form of (h(g(x)))^n, where h(x) = sqrt(x) and g(x) = x + 8, we can use the chain rule, which states that the derivative of (h(g(x)))^n is n * (h(g(x)))^(n-1) * h'(g(x)) * g'(x).

Applying the chain rule:
f'(x) = 1/2 * (x + 8)^(-1/2) * 1

The derivative of g(x) = x^2 + 20x can be found using the power rule and the sum rule.

Applying the power rule:
g'(x) = 2x^(2-1) + 20x^(1-1)
= 2x + 20

Step 3: Plug the results into the product rule formula.

f'(x) = 1/2 * (x + 8)^(-1/2) * 1
g(x) = x^2 + 20x
g'(x) = 2x + 20

f'(x)g(x) + f(x)g'(x) = (1/2 * (x + 8)^(-1/2) * 1)*(x^2 + 20x) + (sqrt(x + 8) * (2x + 20))

Simplifying this expression will give you the final derivative.

It is crucial to remember that when computing derivatives, double-check your work and be cautious of any potential errors. Good luck on your quiz!