Question

Differentiate the following with respect to x.

x^{sin x}

Answer 1

Let y = x^{sin x}

Taking logarithm on both sides we get,

log y = log(x^{sin x})

log y = sin x log x

Differentiating with respect to x,

`1/y * "dy"/"dx" = sin x "d"/"dx" (log x) + log x "d"/"dx" (sin x)`

`1/y * "dy"/"dx" = sin x (1/x) + log x (cos x)`

`"dy"/"dx" = y[(sin x)/x + cos x log x]`

`therefore "dy"/"dx" = x^(sin x) [(sin x)/x + cos x log x]`