If f(x)=(3x+6)–1, find f'(x).

f'= 3 I am not certain what is here, this is very simple, assuming calculus.

wolfram alpha!

To find the derivative of the function f(x) = (3x + 6)^-1, we can use the power rule for differentiation. The power rule states that if we have a function of the form f(x) = x^n, then its derivative f'(x) is given by f'(x) = nx^(n-1).

In this case, our function f(x) = (3x + 6)^-1 has the form of (u)^-1, where u = 3x + 6.

To find the derivative, we'll need to apply the chain rule. The chain rule states that if we have a composition of functions f(g(x)), then its derivative is given by f'(g(x)) * g'(x).

First, let's find the derivative of the inner function g(x) = 3x + 6. The derivative of g(x) with respect to x is simply the coefficient of x, which is 3.

Now, let's differentiate the outer function f(u) = u^-1. To do this, we'll take the derivative of u^-1 with respect to u and multiply it by the derivative of u with respect to x.

The derivative of u^-1 with respect to u can be found using the power rule again. Since the power is -1, we subtract 1 from the power and multiply it by the coefficient. So, we have -1u^(-1-1) = -u^(-2).

The derivative of u with respect to x is the derivative of 3x + 6, which we found to be 3.

Finally, we multiply these two derivatives together: (-u^(-2)) * (3) = -3/u^2.

So, the derivative of f(x) = (3x + 6)^-1 is f'(x) = -3/(3x + 6)^2.