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

The derivative of (cos(x)^{sin(x)}) is cos^{sin(x)}(x)(cos(x)ln(cos(x))-tan(x)sin(x))

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

The first derivative of (cos(x)^{sin(x)}) is cos^{sin(x)}(x)(cos(x)ln(cos(x))-tan(x)sin(x))

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

cos^{s i n (x)} (x) (cos (x) ln (cos (x)) - tan (x) sin (x))

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

= e^{s i n (x) l n (c o s (x))} (cos (x) ln (cos (x)) - tan (x) sin (x))

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

= cos^{s i n (x)} (x) (cos (x) ln (cos (x)) - tan (x) sin (x))

e^{x}, 14 e^{- x} + 5, x = 0

\int x (4 x^{2} + 8) d x

\int x 1 - x^{4} d x

f (x) = 2 x, at x = 8

\int_{0}^{1} 2 (x^{2} + 1) e^{- x} d x

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