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प्रश्न
Differentiate the following function with respect to x:
`(cos x)^x; ("where" x in (0, pi/2))`
बेरीज
उत्तर
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)`
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