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

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

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

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

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

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

f (x) = e^{x - 3}

f (x) = 2 x^{2} + x

y = sin^{4} (x)

f (x) = x^{2} - 7 x + 8

f (x) = 3 + 2 x

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