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

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

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

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

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\frac{d}{d x} ((ln (x))^{c o s (x)})

ln^{c o s (x)} (x) (- sin (x) ln (ln (x)) + \frac{cos ( x )}{x ln ( x )})

Solution steps

\frac{d}{d x} ((ln (x))^{c o s (x)})

Apply exponent rule:

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

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

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

Apply the chain rule:

e^{c o s (x) l n (l n (x))} \frac{d}{d x} (cos (x) ln (ln (x)))

= e^{c o s (x) l n (l n (x))} \frac{d}{d x} (cos (x) ln (ln (x)))

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

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

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

= ln^{c o s (x)} (x) (- sin (x) ln (ln (x)) + \frac{cos ( x )}{x ln ( x )})

x \to 3 - lim (\frac{[ x ] - x}{3 - x})

\int \frac{1}{4 + 9 x ^{2}} d x

\frac{d}{d x} (3 x (8 x^{3} - 2))

\frac{d}{d x} (e^{2 l n (x)})

\frac{d x}{d t} + 4 t^{3} x^{5} + \frac{x}{t} = 0

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