Question

Differentiate the following function with respect to x:

`(cos x)^x; ("where"  x in (0,  pi/2))`

Answer 1

Let y = (cos x)^x, then y = `e^(xlog_ecosx)`

On differentiating both sides with respect to x, we get

`dy/dx = e^(xlog_ecosx) d/dx(xlog_ecosx)`

⇒ `dy/dx = (cosx)^x{log_ecosxd/dx(x) + xd/dx(log_ecosx)}`

⇒ `dy/dx = (cosx)^x {log_e cosx + x*1/(cosx) (-sinx)}`

⇒ `dy/dx = (cosx)^x(log_ecosx - xtanx)`