What is the derivative of (cos(6x)^x) ?

The derivative of (cos(6x)^x) is cos^x(6x)(ln(cos(6x))-6xtan(6x))

What is the first derivative of (cos(6x)^x) ?

The first derivative of (cos(6x)^x) is cos^x(6x)(ln(cos(6x))-6xtan(6x))

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\frac{d}{d x} ((cos (6 x))^{x})

cos^{x} (6 x) (ln (cos (6 x)) - 6 x tan (6 x))

Solution steps

\frac{d}{d x} ((cos (6 x))^{x})

Apply exponent rule:

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

(cos (6 x))^{x} = e^{x l n (c o s (6 x))}

= \frac{d}{d x} (e^{x l n (c o s (6 x))})

Apply the chain rule:

e^{x l n (c o s (6 x))} \frac{d}{d x} (x ln (cos (6 x)))

= e^{x l n (c o s (6 x))} \frac{d}{d x} (x ln (cos (6 x)))

\frac{d}{d x} (x ln (cos (6 x))) = ln (cos (6 x)) - 6 x tan (6 x)

= e^{x l n (c o s (6 x))} (ln (cos (6 x)) - 6 x tan (6 x))

e^{x l n (c o s (6 x))} = cos^{x} (6 x)

= cos^{x} (6 x) (ln (cos (6 x)) - 6 x tan (6 x))

x \to 0 lim (x^{3} ln (x))

x \to 0 lim (\frac{1 - 6 x}{x})

\int \frac{x}{( x + 1 )} d x

\int \frac{6}{x ^{2} + 1} d x

\int \frac{sin ( x )}{1 - cos ( x )} d x

What is the derivative of (cos(6x)^x) ?
The derivative of (cos(6x)^x) is cos^x(6x)(ln(cos(6x))-6xtan(6x))
What is the first derivative of (cos(6x)^x) ?
The first derivative of (cos(6x)^x) is cos^x(6x)(ln(cos(6x))-6xtan(6x))