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

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

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

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

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

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

Solution steps

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

Apply the chain rule:

\frac{1}{x ^{s i n (x)}} \frac{d}{d x} (x^{s i n (x)})

= \frac{1}{x ^{s i n (x)}} \frac{d}{d x} (x^{s i n (x)})

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

= \frac{1}{x ^{s i n (x)}} x^{s i n (x)} (cos (x) ln (x) + \frac{sin ( x )}{x})

Simplify

\frac{1}{x ^{s i n (x)}} x^{s i n (x)} (cos (x) ln (x) + \frac{sin ( x )}{x}) : cos (x) ln (x) + \frac{sin ( x )}{x}

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

\int \frac{1}{x 9 x - 1} d x

\int (\frac{cos ( x )}{sin ( 2 x )}) d x

\frac{x + 1770}{30 ( x + 2 )}

\frac{d}{d x} (2 x (3 x - 2)^{4})

\int \frac{t ^{5}}{t ^{6} + 1} d t

What is the derivative of ln(x^{sin(x})) ?
The derivative of ln(x^{sin(x})) is cos(x)ln(x)+(sin(x))/x
What is the first derivative of ln(x^{sin(x})) ?
The first derivative of ln(x^{sin(x})) is cos(x)ln(x)+(sin(x))/x