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

The derivative of sin^{e^x}(x) is sin^{e^x}(x)(e^xln(sin(x))+e^xcot(x))

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

The first derivative of sin^{e^x}(x) is sin^{e^x}(x)(e^xln(sin(x))+e^xcot(x))

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\frac{d}{d x} (sin^{e^{x}} (x))

sin^{e^{x}} (x) (e^{x} ln (sin (x)) + e^{x} cot (x))

Solution steps

\frac{d}{d x} (sin^{e^{x}} (x))

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

\frac{d}{d x} (e^{x} ln (sin (x))) = e^{x} ln (sin (x)) + cot (x) e^{x}

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

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

= sin^{e^{x}} (x) (e^{x} ln (sin (x)) + e^{x} cot (x))

\int - \frac{1300}{x ^{2}} d x

\int (x + 1)^{2} (1 - x)^{5} d x

e^{c o s (3 x^{2} + x)}

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

\frac{d}{d x} (\frac{3 x}{x})

What is the derivative of sin^{e^x}(x) ?
The derivative of sin^{e^x}(x) is sin^{e^x}(x)(e^xln(sin(x))+e^xcot(x))
What is the first derivative of sin^{e^x}(x) ?
The first derivative of sin^{e^x}(x) is sin^{e^x}(x)(e^xln(sin(x))+e^xcot(x))