## derivative of natural logs, f(x)= ln(x^3)

### To find the derivative of the natural logarithm function, f(x) = ln(x^3), we can apply the chain rule

To find the derivative of the natural logarithm function, f(x) = ln(x^3), we can apply the chain rule.

The chain rule states that if we have a composition of functions, for example f(g(x)), the derivative can be calculated by multiplying the derivative of the outer function (f'(g(x))) with the derivative of the inner function (g'(x)).

In this case, we can rewrite f(x) = ln(x^3) as f(x) = ln(u) where u = x^3. To find the derivative, we will differentiate the natural logarithm function with respect to u and then multiply it by the derivative of u with respect to x.

Let’s start by finding the derivative of the natural logarithm function, ln(u), with respect to u. We know that the derivative of ln(u) is 1/u. Therefore, d(ln(u))/du = 1/u.

Next, we find the derivative of u = x^3 with respect to x. Applying the power rule, we differentiate each term of x^3 separately. The power rule states that d(x^n)/dx = n*x^(n-1). Therefore, d(u)/dx = 3*x^(3-1).

Now, we can use the chain rule and multiply the derivatives we found:

f'(x) = d(ln(u))/dx = (d(ln(u))/du) * (du/dx)

= (1/u) * (3*x^(3-1))

= (1/x^3) * (3*x^2)

= 3x^2 / x^3

= 3/x

Therefore, the derivative of f(x) = ln(x^3) with respect to x is f'(x) = 3/x.

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