Question 6: Differentiate the function with respect to x. (x+1/x )^x + x^(1+1/x)
Class 12 - Math - Continuity and Differentiability
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Refer to the attachment.
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Formulas used :-
• log xⁿ = nlogx
• d/dx(log x) = 1/x
•d/dx(xⁿ) = nxⁿ-¹
×××Logarithmic Differentiation×××
----------------------------
Formulas used :-
• log xⁿ = nlogx
• d/dx(log x) = 1/x
•d/dx(xⁿ) = nxⁿ-¹
×××Logarithmic Differentiation×××
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★ DIFFERENTIABILITY ★
Given , y = (x + 1/x )^x + x ^ 1 + 1/x
Let it be dy/dx = du/dx + dv/dx
Then , log u = x log ( x + 1/x )
log v = 1 + 1/x ( log x )
du/dx = u [log ( x + 1/x ) + x ( 1 - 1/x² ) ]
dv/dx = v [ 1/x² log x + 1 + 1/x / x ]
Therefore ,
dy/dx = x + 1/x ^ x [ x² - 1 / x² + 1 + log ( x + 1/x ) ] + x ^ 1 + 1/x [ x + 1 - log x / x² ]
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Given , y = (x + 1/x )^x + x ^ 1 + 1/x
Let it be dy/dx = du/dx + dv/dx
Then , log u = x log ( x + 1/x )
log v = 1 + 1/x ( log x )
du/dx = u [log ( x + 1/x ) + x ( 1 - 1/x² ) ]
dv/dx = v [ 1/x² log x + 1 + 1/x / x ]
Therefore ,
dy/dx = x + 1/x ^ x [ x² - 1 / x² + 1 + log ( x + 1/x ) ] + x ^ 1 + 1/x [ x + 1 - log x / x² ]
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