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

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

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

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

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

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

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

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

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

x \to 0 - lim (x^{3} + 3 x + 2)

\frac{d}{d x} (\frac{x - 6}{7 x ^{2} e ^{x}})

e^{x} y \cdot \frac{d y}{d x} = e^{- y} + e^{- 3 x - y}

x \to 0 + lim ((tan (2 x))^{2})

\frac{d}{d z} (2 x y)

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