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

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

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

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

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

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

Solution steps

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

Apply exponent rule:

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

cos^{s i n (x)} (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))

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