What is the derivative of (cos(5x)^x) ?

The derivative of (cos(5x)^x) is cos^x(5x)(ln(cos(5x))-5xtan(5x))

What is the first derivative of (cos(5x)^x) ?

The first derivative of (cos(5x)^x) is cos^x(5x)(ln(cos(5x))-5xtan(5x))

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\frac{d}{d x} ((cos (5 x))^{x})

cos^{x} (5 x) (ln (cos (5 x)) - 5 x tan (5 x))

Solution steps

\frac{d}{d x} ((cos (5 x))^{x})

Apply exponent rule:

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

(cos (5 x))^{x} = e^{x l n (c o s (5 x))}

= \frac{d}{d x} (e^{x l n (c o s (5 x))})

Apply the chain rule:

e^{x l n (c o s (5 x))} \frac{d}{d x} (x ln (cos (5 x)))

= e^{x l n (c o s (5 x))} \frac{d}{d x} (x ln (cos (5 x)))

\frac{d}{d x} (x ln (cos (5 x))) = ln (cos (5 x)) - 5 x tan (5 x)

= e^{x l n (c o s (5 x))} (ln (cos (5 x)) - 5 x tan (5 x))

e^{x l n (c o s (5 x))} = cos^{x} (5 x)

= cos^{x} (5 x) (ln (cos (5 x)) - 5 x tan (5 x))

\int \frac{1}{16 x ^{2} + 1} d x

tan^{2} (x) + 1

y e^{x}

y^{^{'}} = y (5 - y)

\int x^{- 2} e^{x} d x

What is the derivative of (cos(5x)^x) ?
The derivative of (cos(5x)^x) is cos^x(5x)(ln(cos(5x))-5xtan(5x))
What is the first derivative of (cos(5x)^x) ?
The first derivative of (cos(5x)^x) is cos^x(5x)(ln(cos(5x))-5xtan(5x))