If y=x^{sin x} +sin(x^x), find

OR

If find

Let u = x^{sin x}

Taking log both sides,

⇒ log u = sin x log x

On differentiating,

Let, v = sin (x^x)

Let, t = x^x

⇒ log t = x log x

On differentiating,

Using (2),

As, y = u + v

So,

From (1) and (3),

OR

y = log (1 + 2t² + t⁴)

Applying the formula (a + b)² = a² + b² + 2ab in (1 + 2t² + t⁴).

⇒ y = log (1+t²)²

As log xⁿ = n log x

⇒ y = 2 log (1+t²)

On differentiating,

x = tan^-1 t

On differentiating,

Now,

Again differentiating,

=4(1+t²)

Therefore,