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

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

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

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

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

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

\frac{d y}{d t} = \frac{y}{t + 1} + 4 t^{2} + 4 t, y (1) = 6

\frac{d}{d x} ((A x + B) e^{2 x})

x \to 3 lim (5 x^{2} - 7 x - 3)

(10 x^{2} - 20 x + 20) e^{- 6 x}

\int \frac{( sin ^{3} ( x ) )}{cos ( x )} d x

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