I f space y space equals space x to the power of x space p r o v e space t h a t fraction numerator space d squared y over denominator d x squared end fraction minus 1 over y open parentheses fraction numerator d y over denominator d x end fraction close parentheses squared space minus fraction numerator space y over denominator x end fraction space equals space 0

Asked by inderjeet.saluja | 24th Aug, 2014, 05:02: PM

Expert Answer:

y equals x to the power of x  fraction numerator d y over denominator d x end fraction equals x to the power of x open parentheses ln space x plus 1 close parentheses  y over x open parentheses fraction numerator d y over denominator d x end fraction close parentheses equals x to the power of x minus 1 end exponent cross times x to the power of x open parentheses ln space x plus 1 close parentheses space minus minus minus minus minus minus minus 1  open parentheses fraction numerator d y over denominator d x end fraction close parentheses squared equals open square brackets x to the power of x open parentheses ln space x plus 1 close parentheses close square brackets squared equals x to the power of 2 x end exponent open parentheses ln space x plus 1 close parentheses squared  open parentheses fraction numerator d y over denominator d x end fraction close parentheses squared equals y. x to the power of x. open parentheses ln space x plus 1 close parentheses squared  1 over y open parentheses fraction numerator d y over denominator d x end fraction close parentheses squared equals x to the power of x. open parentheses ln space x plus 1 close parentheses squared space minus minus minus minus minus minus minus minus minus minus minus 2  fraction numerator d over denominator d x end fraction open parentheses fraction numerator d y over denominator d x end fraction close parentheses equals fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator d over denominator d x end fraction open parentheses x to the power of x open parentheses ln space x plus 1 close parentheses close parentheses equals fraction numerator d over denominator d x end fraction open parentheses x to the power of x space l n space x close parentheses plus fraction numerator d over denominator d x end fraction open square brackets x to the power of x close square brackets  fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator d over denominator d x end fraction open parentheses x to the power of x space l n space x close parentheses plus x to the power of x open parentheses ln space x plus 1 close parentheses  fraction numerator d squared y over denominator d x squared end fraction equals open square brackets ln space x space open parentheses x to the power of x open parentheses ln space x plus 1 close parentheses close parentheses plus x to the power of x cross times 1 over x close square brackets plus x to the power of x open parentheses ln space x plus 1 close parentheses  fraction numerator d squared y over denominator d x squared end fraction equals open parentheses x to the power of x open parentheses ln space x plus 1 close parentheses close parentheses open parentheses 1 plus ln space x close parentheses plus x to the power of 2 x minus 1 end exponent open parentheses ln space x plus 1 close parentheses  f r o m space 1 space a n d space 2 comma space s u b s t i t u t e space t h e space v a l u e s space i n space a b o v e space e q u a t i o n  fraction numerator d squared y over denominator d x squared end fraction equals 1 over y open parentheses fraction numerator d y over denominator d x end fraction close parentheses squared plus y over x open parentheses fraction numerator d y over denominator d x end fraction close parentheses  fraction numerator d squared y over denominator d x squared end fraction minus 1 over y open parentheses fraction numerator d y over denominator d x end fraction close parentheses squared minus y over x open parentheses fraction numerator d y over denominator d x end fraction close parentheses equals 0

Answered by Dharma Teja | 25th Aug, 2014, 11:14: AM

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