Expert Answers

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The function f(x) = x^(sin x)

Let y = f(x) = x^(sin x)

Take the natural log of both the sides

ln y = ln [ x^(sin x)]

=> ln y = sin x * ln x

Differentiate both the sides with respect to x

=> (1/y)(dy/dx) = (sin x)/x + cos x * ln x

=> dy/dx = y*[sin x + (cos x)(ln x)*x]/x

=> dy/dx = [(x^(sin x))*(sin x) + x^(sin x)(cos x)(ln x)*x]/x

=> dy/dx = [(x^(sin x))*(sin x) + x^(sin x + 1)(cos x)(ln x)]/x

f'(x) = [(x^(sin x))*(sin x) + x^(sin x + 1)(cos x)(ln x)]/x

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