What is the derivative of (tan(x)^{cos(x)}) ?

The derivative of (tan(x)^{cos(x)}) is tan^{cos(x)}(x)(-sin(x)ln(tan(x))+2csc(2x)cos(x))

What is the first derivative of (tan(x)^{cos(x)}) ?

The first derivative of (tan(x)^{cos(x)}) is tan^{cos(x)}(x)(-sin(x)ln(tan(x))+2csc(2x)cos(x))

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\frac{d}{d x} ((tan (x))^{c o s (x)})

tan^{c o s (x)} (x) (- sin (x) ln (tan (x)) + 2 csc (2 x) cos (x))

Solution steps

\frac{d}{d x} ((tan (x))^{c o s (x)})

Apply exponent rule:

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

(tan (x))^{c o s (x)} = e^{c o s (x) l n (t a n (x))}

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

Apply the chain rule:

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

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

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

= e^{c o s (x) l n (t a n (x))} (- sin (x) ln (tan (x)) + 2 csc (2 x) cos (x))

e^{c o s (x) l n (t a n (x))} = tan^{c o s (x)} (x)

= tan^{c o s (x)} (x) (- sin (x) ln (tan (x)) + 2 csc (2 x) cos (x))

What is the derivative of (tan(x)^{cos(x)}) ?
The derivative of (tan(x)^{cos(x)}) is tan^{cos(x)}(x)(-sin(x)ln(tan(x))+2csc(2x)cos(x))
What is the first derivative of (tan(x)^{cos(x)}) ?
The first derivative of (tan(x)^{cos(x)}) is tan^{cos(x)}(x)(-sin(x)ln(tan(x))+2csc(2x)cos(x))