Chain rule with a power of ln(x)

Let $g(x) = (\ln(x))^3$ . Find $g'(x)$ .

Hint

Treat $g(x)$ as an outer power function raised to the 3rd power applied to an inner function $u = \ln(x)$ . Use the chain rule: $g'(x) = 3(\ln(x))^2 \cdot \dfrac{d}{dx}[\ln(x)]$ .

Step-by-step solution

Identify the outer function as $f(u) = u^3$ and the inner function as $u = \ln(x)$ , so $g(x) = f(u(x))$ . By the chain rule, $g'(x) = f'(u(x)) \cdot u'(x) = 3(\ln(x))^2 \cdot u'(x)$ .

Since $u(x) = \ln(x)$ , its derivative is $u'(x) = \dfrac{1}{x}$ . Substituting this in gives $g'(x) = 3(\ln(x))^2 \cdot \dfrac{1}{x} = \dfrac{3(\ln(x))^2}{x}$ .

Chain rule with a power of ln(x)

Related problems