Derivative of a General Power Function

Derivative of a General Power Function

For:

$$y=u(x)^v(x)$$

logarithmic differentiation gives:

$$\frac{y'}{y}=v'(x)\ln u(x)+v(x)\frac{u'(x)}{u(x)}$$

Hence:

$$y'=u(x)^{v(x)}\left[v'(x)\ln u(x)+v(x)\frac{u'(x)}{u(x)}\right]$$

Usage

- Used when both the base and exponent depend on $$x$$.