Question

Differentiate (log x)^{log x}, x > 1 w.r.t.x.

Name: Wired Faculty - The Best Learning App
Rating: 4.6 (1864 reviews)

Solution

Let space straight y equals left parenthesis log space straight x right parenthesis to the power of log space straight x end exponent therefore space log space straight y equals log left parenthesis log space straight x right parenthesis to the power of log space straight x end exponent equals log space straight x. log left parenthesis log space straight x right parenthesis therefore space 1 over straight y dy over dx equals log space straight x. fraction numerator 1 over denominator log space straight x end fraction.1 over straight x plus log left parenthesis log space straight x right parenthesis.1 over straight x equals 1 over straight x plus fraction numerator log left parenthesis log space straight x right parenthesis over denominator straight x end fraction therefore space space space space space space dy over dx equals straight y open square brackets fraction numerator 1 plus log left parenthesis log space straight x right parenthesis over denominator straight x end fraction close square brackets therefore space space space space space fraction numerator dy over denominator space dx end fraction equals left parenthesis log space straight x right parenthesis to the power of log space straight x end exponent open square brackets fraction numerator 1 plus log left parenthesis log space straight x right parenthesis over denominator straight x end fraction close square brackets

For Daily Free Study Material Join wiredfaculty