Differentiate given problems w.r.t.x.
$(s i n x - c o s x)^{(} s i n x - c o s x)$

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Solution

Let y= $(s i n x - c o s x)^{(s i n x - c o s x)}$

Taking log on both sides w.r.t.x, we get

log y = $l o g (s i n x - c o s x)^{(s i n x - c o s x)}$

or $l o g y = (s i n x - c o s x) l o g (s i n x - c o s x) (∴ l o g m^{n} = n l o g . m)$

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

$\frac{1}{y} \frac{d y}{d x} = (s i n x - c o s x) \frac{c o s x + s i n x}{s i n x - c o s x} + [l o g (s i n x - c o s x)] (c o s x + s i n x)$

$\Rightarrow \frac{d y}{d x} = y (c o s x + s i n x) [1 + l o g (s i n x - c o s x)]$

= $(c o s x + s i n x) (s i n x - c o s x)^{(} s i n x - c o s x) 1 + l o g (s i n x - c o s x)$

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