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Question
Differentiate w.r.t. x the function:
`(sin x - cos x)^(sin x - cos x), pi/4 < x < (3pi)/4`
Solution
Let, y = `(sin x- cos x)^(sin x – cos x)`
Taking logarithm on both sides,
log y = log (sin x – cosx)(sin x – cos x)
log y=(sin x – cos x)log (sin x – cosx), [∵ log mn = n log m]
On differentiating with respect to x,
`1/y dy/dx = (sin x - cos x) d/dx log (sin x - cos x) + log (sin x - cos x) d/dx (sin x - cos x)`
`= (sin x - cos x) xx 1/(sin x - cos x) d/dx (sin x - cos x) + log (sin x - cos x)(cos x + sin x)`
`= (cos x + sin x)[1 + log (sin x - cos x)]`
`therefore "dy"/"dx" = "y" (cos x + sin x)[1 + log (sin x - x)]`
`= (sin x - cos x)^(sin x - cos x) (cos x + sin x)[1 + log (sin x- cos x)], sin x > cos x`
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