Find the derivative using the product rule.
{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.
Change minus to plus and then it's all good.
To find the derivative using the product rule, you need to differentiate each term separately and then combine them using a specific formula.
Let's name the first term as f(x) = √(x + 8) and the second term as g(x) = x^2 + 20x.
To apply the product rule, you need to follow these steps:
1. Differentiate the first term, f(x), with respect to x.
- The derivative of √(x + 8) can be found using the chain rule, which states that d(u^n)/dx = n(u^(n-1))(du/dx), where u is a function of x.
- Here, u = x + 8 and n = 1/2.
- Applying the chain rule, we get d(√(x + 8))/dx = (1/2)(x + 8)^(-1/2)(1),
which simplifies to (1/2√(x + 8)).
2. Differentiate the second term, g(x), with respect to x.
- The derivative of x^2 + 20x can be found using the power rule, which states that d(x^n)/dx = nx^(n-1), where n is a constant.
- Applying the power rule, we get d(x^2 + 20x)/dx = 2x + 20.
3. Apply the product rule formula to combine the derivatives.
- The product rule states that d(f(x)g(x))/dx = f'(x)g(x) + f(x)g'(x).
- Plugging in the derivatives we found: (1/2√(x + 8))(x^2 + 20x) + (√(x + 8))(2x + 20).
Simplifying the expression gives:
(x^2 + 20x)/(2√(x + 8)) + (2x + 20)√(x + 8).
This is the derivative of the given function using the product rule. Make sure to simplify the expression further or follow any additional instructions given in the question.