Logarithm 10

Calculus Level 2

Given the function $f(u) = \ln(u^a b^u)$ , where $\ln$ denotes the natural logarithm, find the first derivative $f'(u)$ .

\frac{a b}{u} + \ln (a)

\frac{b}{u} - u^2 \ln (a)

\frac{a}{u} + u \ln (b)

\frac{a}{u} + \ln (b)

\frac{ab}{u} + a \ln (b)

1 solution

Tom Engelsman
Dec 16, 2020

The above function can be rewritten as:

$f(u) = \ln(u^{a} b^{u}) = \ln(u^a) + \ln(b^u) = a \cdot \ln(u) + u \cdot \ln(b)$

which has the first derivative $\boxed{f'(u) = \frac{a}{u} + \ln(b) }.$