What is the derivative of y=(cos(x))^x ?

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

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

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

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derivative

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

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