What is the derivative of (cos(2x)^x) ?

The derivative of (cos(2x)^x) is cos^x(2x)(ln(cos(2x))-2xtan(2x))

What is the first derivative of (cos(2x)^x) ?

The first derivative of (cos(2x)^x) is cos^x(2x)(ln(cos(2x))-2xtan(2x))

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\frac{d}{d x} ((cos (2 x))^{x})

cos^{x} (2 x) (ln (cos (2 x)) - 2 x tan (2 x))

Solution steps

\frac{d}{d x} ((cos (2 x))^{x})

Apply exponent rule:

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

(cos (2 x))^{x} = e^{x l n (c o s (2 x))}

= \frac{d}{d x} (e^{x l n (c o s (2 x))})

Apply the chain rule:

e^{x l n (c o s (2 x))} \frac{d}{d x} (x ln (cos (2 x)))

= e^{x l n (c o s (2 x))} \frac{d}{d x} (x ln (cos (2 x)))

\frac{d}{d x} (x ln (cos (2 x))) = ln (cos (2 x)) - 2 x tan (2 x)

= e^{x l n (c o s (2 x))} (ln (cos (2 x)) - 2 x tan (2 x))

e^{x l n (c o s (2 x))} = cos^{x} (2 x)

= cos^{x} (2 x) (ln (cos (2 x)) - 2 x tan (2 x))

\int_{1}^{\infty} 2 e^{- 2 x} d x

13 x^{3}

\int (x^{2}) arcsin (x) d x

\frac{\partial}{\partial y} (\frac{4 x - 4 z}{y + 4 z})

\int (2 y^{2}) d y

What is the derivative of (cos(2x)^x) ?
The derivative of (cos(2x)^x) is cos^x(2x)(ln(cos(2x))-2xtan(2x))
What is the first derivative of (cos(2x)^x) ?
The first derivative of (cos(2x)^x) is cos^x(2x)(ln(cos(2x))-2xtan(2x))