Question

Differentiate the function with respect to x:

`(sin x - cos x)^((sin x - cos x)), pi/4 < x < (3pi)/4`

Answer 1

Let, y = `(sin x- cos x)^((sin x – cos x))`

Taking logarithm on both sides,

log y = log (sin x – cos x) (sin x – cos x)

log y = (sin x – cos x) log (sin x – cos x)   ...[∵ 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)]

∴ `dy/dx` = y (cos x + sin x) [1 + log (sin x − cos x)]

= `(sin x - cos x)^((sin x - cos x)) (cos x + sin x)[1 + log (sin x- cos x)]`, sin x > cos x