Find the derivative of y = (ln(3x))^4
dy/dx = 3(ln(3x))^3 (1/(3x))(3)
= (3/x)(ln(3x))^3
To find the derivative of y = (ln(3x))^4, we can use the chain rule. The chain rule states that if we have a function of the form f(g(x)), then its derivative is given by f'(g(x)) * g'(x).
Let's break down the function step by step:
Step 1: Start by applying the power rule. The power rule states that if we have a function of the form f(x)^n, then its derivative is given by n*f(x)^(n-1). In this case, we have (ln(3x))^4, so applying the power rule, we get:
dy/dx = 4(ln(3x))^3 * d/dx(ln(3x))
Step 2: Now, we need to find the derivative of ln(3x). The derivative of ln(u) with respect to x is given by du/dx divided by u. In this case, our u is 3x. So:
d/dx(ln(3x)) = (1/(3x)) * d/dx(3x) = (1/(3x)) * 3
Step 3: Simplifying further, we have:
d/dx(ln(3x)) = 1/x
Now, substituting this back into our original equation, we have:
dy/dx = 4(ln(3x))^3 * (1/(3x)) * 3
Simplifying, we get:
dy/dx = 12(ln(3x))^3 / (3x)
Therefore, the derivative of y = (ln(3x))^4 is dy/dx = 12(ln(3x))^3 / (3x).