Question

(1+1/x)^{x}+ x^{1+\frac{1}{x} प्रदत्त फलनों का x के सापेक्ष अवकलन कीजिए

Answer 1

$dy/dx = (1 + 1/x)^x ( -1/(x + 1) + log(1 + 1/x)) + x^{1+\frac{1}{x}} \times (x + 1 - logx)/x^{2}$

Step-by-step explanation:

$y = (1+1/x)^{x}+ x^{1+\frac{1}{x}$

y = u + v

dy/dx = du/dx  + dv/dx

u  = (1 + 1/x)ˣ

log u  = x log (1  + 1/x)

(1/u) du/dx   = (x /(1 + 1/x) )  (-1/x²)  + log(1 + 1/x)

=> (1/u) du/dx   =  -1/(x + 1) +  log(1 + 1/x)

=>du/dx  = u (  -1/(x + 1) +  log(1 + 1/x))

=> du/dx  = (1 + 1/x)ˣ (  -1/(x + 1) +  log(1 + 1/x))

$v = x^{1+\frac{1}{x}$

log v  =  (1 + 1/x)  logx

=> 1/v  dv/dx  = (1 + 1/x) (1/x)  +  ( -1/x²)logx

=>  1/v  dv/dx =  (x + 1 - logx)/x²

=> dv/dx  = v  (x + 1 - logx)/x²

$dv/dx = x^{1+\frac{1}{x}} \times (x + 1 - logx)/x^{2}$

$dy/dx = (1 + 1/x)^x ( -1/(x + 1) + log(1 + 1/x)) + x^{1+\frac{1}{x}} \times (x + 1 - logx)/x^{2}$

और अधिक जानें :

sin(x²+5)"

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sin (ax+b) फलन का अवकलन कीजिए

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