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CBSE Class 12-science Answered

Differentiate (logx)(raised to)X + X(raised to)logx w.r.t logx(raised to)X for X>0
Asked by pratib1968 | 26 Nov, 2018, 10:59: AM
answered-by-expert Expert Answer
begin mathsize 12px style L e t space V left parenthesis x right parenthesis space equals space open parentheses log x close parentheses to the power of x space plus space x to the power of log x end exponent space equals space U left parenthesis x right parenthesis space plus space W left parenthesis x right parenthesis w h e r e space U left parenthesis x right parenthesis space equals space open parentheses log space x close parentheses to the power of x space space space space a n d space space space W left parenthesis x right parenthesis space equals space x to the power of log x end exponent log space left square bracket space U left parenthesis x right parenthesis space right square bracket space equals space x space log left square bracket space log x space right square bracket space space fraction numerator 1 over denominator U left parenthesis x right parenthesis end fraction fraction numerator d U left parenthesis x right parenthesis over denominator d x end fraction space equals space log space left square bracket space log x space right square bracket space plus x space fraction numerator 1 over denominator log x end fraction 1 over x space equals space log space open square brackets log x close square brackets space plus fraction numerator 1 over denominator log x end fraction fraction numerator d U left parenthesis x right parenthesis over denominator d x end fraction space equals space open parentheses log x close parentheses to the power of x space open curly brackets log open square brackets log x close square brackets space plus space fraction numerator 1 over denominator log x end fraction close curly brackets log open square brackets W left parenthesis x right parenthesis close square brackets space equals space log x space log x space equals space open parentheses log x close parentheses squared space space fraction numerator 1 over denominator W left parenthesis x right parenthesis end fraction fraction numerator d W left parenthesis x right parenthesis over denominator d x end fraction space equals space 2 space log x space 1 over x fraction numerator d W left parenthesis x right parenthesis over denominator d x end fraction space equals space x to the power of log x end exponent space open curly brackets 2 space log x space 1 over x close curly brackets space equals space 2 space x to the power of log x minus 1 end exponent space log x fraction numerator d V over denominator d x end fraction space equals space fraction numerator d U left parenthesis x right parenthesis over denominator d x end fraction plus fraction numerator d W left parenthesis x right parenthesis over denominator d x end fraction space equals space open parentheses log x close parentheses to the power of x space open curly brackets log open square brackets log x close square brackets space plus space fraction numerator 1 over denominator log x end fraction close curly brackets space plus space space 2 space x to the power of log x minus 1 end exponent space log x end style
begin mathsize 12px style fraction numerator d V over denominator d x end fraction space equals space fraction numerator d U left parenthesis x right parenthesis over denominator d x end fraction plus fraction numerator d W left parenthesis x right parenthesis over denominator d x end fraction space equals space open parentheses log x close parentheses to the power of x space open curly brackets log open square brackets log x close square brackets space plus space fraction numerator 1 over denominator log x end fraction close curly brackets space plus space space 2 space x to the power of log x minus 1 end exponent space log x fraction numerator d V over denominator d U end fraction space equals space open curly brackets bevelled fraction numerator begin display style fraction numerator d V over denominator d x end fraction end style over denominator begin display style fraction numerator d U over denominator d x end fraction end style end fraction close curly brackets space space equals space 1 plus space open square brackets bevelled fraction numerator begin display style fraction numerator d W over denominator d x end fraction end style over denominator begin display style fraction numerator d U over denominator d x end fraction end style end fraction close square brackets space equals space 1 space plus space open curly brackets fraction numerator 2 space x to the power of log x minus 1 end exponent space log x over denominator open parentheses log x close parentheses to the power of x space open curly brackets log open square brackets log x close square brackets space plus space fraction numerator 1 over denominator log x end fraction close curly brackets end fraction close curly brackets end style
Answered by Thiyagarajan K | 26 Nov, 2018, 10:50: PM

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