What is the derivative of y=x^{sin(x)} ?

The derivative of y=x^{sin(x)} is x^{sin(x)}(cos(x)ln(x)+(sin(x))/x)

What is the first derivative of y=x^{sin(x)} ?

The first derivative of y=x^{sin(x)} is x^{sin(x)}(cos(x)ln(x)+(sin(x))/x)

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derivative

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

Solution steps

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

Apply the chain rule:

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

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

\frac{d}{d 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 the derivative of y=x^{sin(x)} ?
The derivative of y=x^{sin(x)} is x^{sin(x)}(cos(x)ln(x)+(sin(x))/x)
What is the first derivative of y=x^{sin(x)} ?
The first derivative of y=x^{sin(x)} is x^{sin(x)}(cos(x)ln(x)+(sin(x))/x)