What is the derivative of f(x) = 10 + sqrt(2x + 32)?

very straightforward question

f(x) = 10 + (2x+32)^(1/2)

f ' (x) = (1/2)(2x+32)^(-1/2) * 2
= 1/√(2x+32)

Why is it multiplied by 2?

To find the derivative of the function f(x) = 10 + sqrt(2x + 32), we can use the power rule and chain rule of differentiation. The power rule states that if we have a function of the form f(x) = x^n, the derivative is given by f'(x) = nx^(n-1).

First, let's rewrite the given function as f(x) = 10 + (2x + 32)^(1/2). Now we can apply the chain rule, which states that if we have a composite function f(g(x)), the derivative is given by f'(g(x)) * g'(x).

Let's define two functions:
g(x) = 2x + 32
h(x) = x^(1/2)

Now we can use the chain rule. The derivative of g(x) with respect to x, g'(x), is simply 2. The derivative of h(x), h'(x), can be found using the power rule, which gives us h'(x) = (1/2)x^(-1/2) = 1 / (2√x).

Applying the chain rule, we have f'(x) = h'(g(x)) * g'(x). Substituting the values, we get f'(x) = (1 / (2√(2x + 32))) * 2.

Simplifying further, we have f'(x) = 1 / √(2x + 32).

Therefore, the derivative of f(x) = 10 + sqrt(2x + 32) is f'(x) = 1 / √(2x + 32).