ln sqrt(X) / x^2
Use the quotient rule:
d/dx [f(x) / g(x)] = [ g(x)*f '(x) - f(x)*g'(x) ] / [g(x)²]
f(x) = ln √x
f '(x) = 1 / 2x
g(x) = x²
g'(x) = 2x
d/dx [f(x) / g(x)]
= [ g(x)*f '(x) - f(x)*g'(x) ] / [g(x)²]
= [ x² * (1 / 2x) - ln (√x) * 2x] / (x²)²
= [ x / 2 - ln (√x) * 2x] / x^4
= [ 1/2 - 2 ln (√x) ] / x³
= [ 1/2 - 2*(1/2)*ln(x) ] / x³
= [ 1/2 - ln(x) ] / x³
1) d(u/v)/dx = (u'v - v'u)/v^2
2 ) d(lnx)/dx = 1/x
3 ) ln (x^1/2) = (1/2) ln (x)
Using 1 & 2 & 3 you get :
((1/2)x - xln(x))/x^4
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Verified answer
Use the quotient rule:
d/dx [f(x) / g(x)] = [ g(x)*f '(x) - f(x)*g'(x) ] / [g(x)²]
f(x) = ln √x
f '(x) = 1 / 2x
g(x) = x²
g'(x) = 2x
d/dx [f(x) / g(x)]
= [ g(x)*f '(x) - f(x)*g'(x) ] / [g(x)²]
= [ x² * (1 / 2x) - ln (√x) * 2x] / (x²)²
= [ x / 2 - ln (√x) * 2x] / x^4
= [ 1/2 - 2 ln (√x) ] / x³
= [ 1/2 - 2*(1/2)*ln(x) ] / x³
= [ 1/2 - ln(x) ] / x³
1) d(u/v)/dx = (u'v - v'u)/v^2
2 ) d(lnx)/dx = 1/x
3 ) ln (x^1/2) = (1/2) ln (x)
Using 1 & 2 & 3 you get :
((1/2)x - xln(x))/x^4