What is the derivative of (cos(3x)^x) ?

The derivative of (cos(3x)^x) is cos^x(3x)(ln(cos(3x))-3xtan(3x))

What is the first derivative of (cos(3x)^x) ?

The first derivative of (cos(3x)^x) is cos^x(3x)(ln(cos(3x))-3xtan(3x))

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

cos^{x} (3 x) (ln (cos (3 x)) - 3 x tan (3 x))

Solution steps

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

Apply exponent rule:

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

(cos (3 x))^{x} = e^{x l n (c o s (3 x))}

= \frac{d}{d x} (e^{x l n (c o s (3 x))})

Apply the chain rule:

e^{x l n (c o s (3 x))} \frac{d}{d x} (x ln (cos (3 x)))

= e^{x l n (c o s (3 x))} \frac{d}{d x} (x ln (cos (3 x)))

\frac{d}{d x} (x ln (cos (3 x))) = ln (cos (3 x)) - 3 x tan (3 x)

= e^{x l n (c o s (3 x))} (ln (cos (3 x)) - 3 x tan (3 x))

e^{x l n (c o s (3 x))} = cos^{x} (3 x)

= cos^{x} (3 x) (ln (cos (3 x)) - 3 x tan (3 x))

f (x) = sin (x) ln (x^{2} + 1)

x \to π lim (9 cot (x))

y = ln (x^{6})

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

What is the derivative of (cos(3x)^x) ?
The derivative of (cos(3x)^x) is cos^x(3x)(ln(cos(3x))-3xtan(3x))
What is the first derivative of (cos(3x)^x) ?
The first derivative of (cos(3x)^x) is cos^x(3x)(ln(cos(3x))-3xtan(3x))