What's the second derivative of $ln (f (x))$ $ln(f(x))$ ?

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Answer 1 · 2016-10-12T15:17:39

$(f(x)f''(x)-(f'(x))^2)/(f(x))^2$

$y=ln(f(x))$

Through the chain rule and the knowledge that $d/dxln(x)=1/x$ , we see that:

$dy/dx=1/f(x)*f'(x)=(f'(x))/f(x)$

To differentiate this again, we will use the quotient rule.

$(d^2y)/(dx^2)=(f(x)d/dxf'(x)-f'(x)d/dxf(x))/(f(x))^2$

$(d^2y)/(dx^2)=(f(x)f''(x)-(f'(x))^2)/(f(x))^2$

What's the second derivative of ln(f(x))ln(f(x))ln(f(x))?