Sponsor Area

Continuity And Differentiability

Question
CBSEENMA12034827

Differentiate the following w.r.t.x: straight x to the power of straight x to the power of straight x end exponent space space space or space straight x to the power of left parenthesis straight x to the power of straight x right parenthesis end exponent

Solution
Let space straight y equals straight x to the power of straight x to the power of straight x end exponent
therefore space log space straight y equals log space straight x to the power of straight x to the power of straight x end exponent equals straight x to the power of straight x log space straight x space space space space space space space space space space space space space space space space rightwards double arrow space log left parenthesis log space straight y right parenthesis equals log left square bracket straight x to the power of straight x. log space straight x right square bracket
therefore space log left parenthesis log space straight y right parenthesis equals log space straight x to the power of straight x plus log left parenthesis log space straight x right parenthesis space space space space space rightwards double arrow space log left parenthesis log space straight y right parenthesis equals straight x. log space straight x plus log left parenthesis log space straight x right parenthesis
Differentiating space straight w. straight r. straight t space straight x comma
fraction numerator 1 over denominator log space straight y end fraction.1 over straight y. dy over dx equals straight x.1 over straight x plus log space straight x.1 plus fraction numerator 1 over denominator log space straight x end fraction.1 over straight x
therefore space dy over dx equals straight y space log space straight y open square brackets 1 plus log space straight x plus fraction numerator 1 over denominator straight x space log space straight x end fraction close square brackets equals straight x to the power of straight x to the power of straight x end exponent. straight x to the power of straight x. log space straight x open square brackets 1 plus log space straight x plus fraction numerator 1 over denominator straight x space log space straight x end fraction close square brackets

Some More Questions From Continuity and Differentiability Chapter