Question

If \[y=[u(x)]^{v(x)}\], which expression gives \[\frac{dy}{dx}\] in logarithmic differentiation?

Options

• \[\frac{dy}{dx}=\frac{1}{y}\left[\frac{v(x)}{u(x)}u'(x)+v'(x)\log[u(x)]\right]\]

• \[\frac{dy}{dx}=y\left[\frac{v(x)}{u(x)}u'(x)+v'(x)\log[u(x)]\right]\]

• \[\frac{dy}{dx}=y\left[\frac{u(x)}{v(x)}v'(x)+u'(x)\log[v(x)]\right]\]

• \[\frac{dy}{dx}=y\left[\frac{v(x)}{u(x)}v'(x)+u'(x)\log[u(x)]\right]\]

Answer 1

For \[y=[u(x)]^{v(x)}\], the derivative contains the factor \[y\]. The first term is \[\frac{v(x)}{u(x)}u'(x)\], and the second is \[v'(x)\log[u(x)]\].