Then log y = x log a Differentiating both sides w.r.t. x , we have dy y dx = log a or dx = y log a Thus = a x log a Alternatively ( log ) = e x log a . log a = a x log a . Example Differentiate x sin x , x > w.r.t.
x . Solution Let y = x sin x . Taking logarithm on both sides, we have log y = sin x log x Therefore . dy y dx = sin (log ) (sin ) or dy y dx = (sin ) x + or − ⋅ Example Find dy dx , if y x + x y + x x = a b .
Solution Given that y x + x y + x x = a b . Putting u = y x , v = x y and w = x x , we get u + v + w = a b Therefore du dv dw ... ( ) Now, u = y x . Taking logarithm on both sides, we have log u = x log y Differentiating both sides w.r.t.
x , we have du u dx (log ) y dx So du x dy x dy u y dx y dx ... ( ) Also v = x y Taking logarithm on both sides, we have log v = y log x Differentiating both sides w.r.t. x , we have dv v dx (log ) So dv v ... ( ) Again w = x x Taking logarithm on both sides, we have log w = x log x .
Differentiating both sides w.r.t. x , we have dw w dx (log ) i.e. dw dx = w ( + log x ) = x x ( + log x ) ... ( ) From ( ), ( ), ( ), ( ), we have x dy y dx + x x ( + log x ) = or ( x .