y = ${\left(\mathit{s}\mathit{i}\mathit{n}\mathit{x}\right)}^{\mathit{x}}+\mathit{s}\mathit{i}{\mathit{n}}^{-1}\sqrt{\mathit{x}}$
Let u = ${\left(\mathit{s}\mathit{i}\mathit{n}\mathit{x}\right)}^{\mathit{x}}$ and v = $\mathit{s}\mathit{i}{\mathit{n}}^{-1}\sqrt{\mathit{x}}$
Now y = u + v
$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{\mathit{d}\mathit{u}}{\mathit{d}\mathit{x}}+\frac{\mathit{d}\mathit{v}}{\mathit{d}\mathit{x}}$ ... (i)
Consider u = ${\left(\mathit{s}\mathit{i}\mathit{n}\mathit{x}\right)}^{\mathit{x}}$
Taking logarithms on both the sides, we have,
logu = xlog (sin x)
Differentiating with respect to x, we have,
$\frac{1}{\mathit{u}}.\frac{\mathit{d}\mathit{u}}{\mathit{d}\mathit{x}}$ = log (sin x) + $\frac{\mathit{x}}{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}$ . cos x
⇒ $\frac{\mathit{d}\mathit{u}}{\mathit{d}\mathit{x}}$ = ${\left(\mathit{s}\mathit{i}\mathit{n}\mathit{x}\right)}^{\mathit{x}}$ (log (sin x) + x cot x) ... (ii)
Consider v = $\mathit{s}\mathit{i}{\mathit{n}}^{-1}\sqrt{\mathit{x}}$
$\frac{\mathit{d}\mathit{v}}{\mathit{d}\mathit{x}}$ = $\frac{1}{\sqrt{1-\mathit{x}}}$ × $\frac{1}{2\sqrt{\mathit{x}}}$ ... (iii)
From (i), (ii) and (iii)
We get , $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = ${\left(\mathit{s}\mathit{i}\mathit{n}\mathit{x}\right)}^{\mathit{x}}$ (log (sinx) + x cot x) + $\frac{1}{2\sqrt{\mathit{x}}\sqrt{1-\mathit{x}}}$