What is (\partial)/(\partial x)(x^{sin(x)}) ?

The answer to (\partial)/(\partial x)(x^{sin(x)}) is x^{sin(x)}(cos(x)ln(x)+(sin(x))/x)

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(\partial)/(\partial x)(x^{sin(x)})

\frac{\partial}{\partial x} (x^{s i n (x)})

x^{s i n (x)} (cos (x) ln (x) + \frac{sin ( x )}{x})

Solution steps

\frac{\partial}{\partial x} (x^{s i n (x)})

Apply exponent rule:

a^{b} = e^{b l n (a)}

x^{s i n (x)} = e^{s i n (x) l n (x)}

= \frac{\partial}{\partial x} (e^{s i n (x) l n (x)})

Apply the chain rule:

e^{s i n (x) l n (x)} \frac{\partial}{\partial x} (sin (x) ln (x))

= e^{s i n (x) l n (x)} \frac{\partial}{\partial x} (sin (x) ln (x))

\frac{\partial}{\partial x} (sin (x) ln (x)) = cos (x) ln (x) + \frac{sin ( x )}{x}

= e^{s i n (x) l n (x)} (cos (x) ln (x) + \frac{sin ( x )}{x})

e^{s i n (x) l n (x)} = x^{s i n (x)}

= x^{s i n (x)} (cos (x) ln (x) + \frac{sin ( x )}{x})

What is (\partial)/(\partial x)(x^{sin(x)}) ?
The answer to (\partial)/(\partial x)(x^{sin(x)}) is x^{sin(x)}(cos(x)ln(x)+(sin(x))/x)