Question

If y = `(sin x)^sin x` , then `"dy"/"dx"` = ?

Options

• `(sin x)^(cos x) {sin log [sin (pi/4 - x/2)]}`

• `(sin x)^(cos x) {sin log [cos (pi/4 + x/2)]}`

• (sin x)^{sin x} [cos x. log (sin x) - cos x]

• (sin x)^{sin x} [cos x. log (sin x) + cos x]

Answer 1

(sin x)^{sin x} [cos x. log (sin x) + cos x]

Explanation:

y = `(sin x)^sin x`

Taking logarithm on both sides, we get

log y = sin x log (sin x)

Differentiating both sides w.r.t. x, we get

`1/y * "dy"/"dx" = sin x * 1/(sin x) * cos x + log (sin x) * cos x`

`=> 1/y * "dy"/"dx" = cos x + cos x log (sin x)`

`=> "dy"/"dx" = (sin x)^(sin x) [cos x + cos x  log (sin x)]`