Question

Darivative of tan x raise to sin x?

Answer 1

`(dy)/(dx) = (tanx)^sinx * ( cosx log(tanx) + secx)`

Explanation:

We are trying to find `(dy)/(dx)` of `y = (tan x)^sinx`

Taking logs ( base `e` )

`logy = log (tanx)^sinx`

Using our laws...

`logy = sinx log (tanx)`

Now we can differentiate implicitly

`1/y (dy)/(dx) = cosxlog(tanx ) + sinx * sec^2 x /tanx`

`sinx * sec^2 x / tanx = sinx * 1/cos^2 x * cosx/sinx`

`= 1/cosx = sec x`

`=> 1/y (dy)/(dx) = cosx log(tanx) + secx`

`=> (dy)/(dx) =y * ( cosx log(tanx) + secx)`

`=> (dy)/(dx) = (tanx)^sinx * ( cosx log(tanx) + secx)`