Question

If `y = (cos x)^x + tan^-1 sqrt(x)`, find `dy/dx`.

Answer 1

Given that `y = (cos x)^x + tan^-1 sqrt(x)`

`y = e^(xlog(cosx)) + tan^-1 sqrt(x)`   ...[Because e^logx = x]

⇒ `dy/dx = e^(xlog(cosx)) {x((-sinx)/(cosx)) + log cos x xx 1} + 1/(1 + x) xx 1/(2sqrt(x))`

Then, `dy/dx = (cosx)^x {logcosx - xtanx} + 1/((1 + x)) xx 1/(2sqrt(x))`