What is the derivative of (cos(9x)^x) ?

The derivative of (cos(9x)^x) is cos^x(9x)(ln(cos(9x))-9xtan(9x))

What is the first derivative of (cos(9x)^x) ?

The first derivative of (cos(9x)^x) is cos^x(9x)(ln(cos(9x))-9xtan(9x))

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\frac{d}{d x} ((cos (9 x))^{x})

cos^{x} (9 x) (ln (cos (9 x)) - 9 x tan (9 x))

Solution steps

\frac{d}{d x} ((cos (9 x))^{x})

Apply exponent rule:

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

(cos (9 x))^{x} = e^{x l n (c o s (9 x))}

= \frac{d}{d x} (e^{x l n (c o s (9 x))})

Apply the chain rule:

e^{x l n (c o s (9 x))} \frac{d}{d x} (x ln (cos (9 x)))

= e^{x l n (c o s (9 x))} \frac{d}{d x} (x ln (cos (9 x)))

\frac{d}{d x} (x ln (cos (9 x))) = ln (cos (9 x)) - 9 x tan (9 x)

= e^{x l n (c o s (9 x))} (ln (cos (9 x)) - 9 x tan (9 x))

e^{x l n (c o s (9 x))} = cos^{x} (9 x)

= cos^{x} (9 x) (ln (cos (9 x)) - 9 x tan (9 x))

\int \frac{x + 1 - 1}{x + 1 + 1} d x

y = 1 9^{x}

\int \frac{2 y}{y ^{2} + 1} d y

x \to 0 - lim (\frac{2}{3} x^{- \frac{1}{3}})

\int_{0}^{\infty} e^{- 14 x} d x

What is the derivative of (cos(9x)^x) ?
The derivative of (cos(9x)^x) is cos^x(9x)(ln(cos(9x))-9xtan(9x))
What is the first derivative of (cos(9x)^x) ?
The first derivative of (cos(9x)^x) is cos^x(9x)(ln(cos(9x))-9xtan(9x))