What is the second derivative of f(x) = ln (x-x^2) ?
1 Answer
Feb 17, 2016
Explanation:
To differentiate the natural logarithm function, use the chain rule. The chain rule states that
d/dx(f(g(x)))=f'(g(x))*g'(x)
Knowing that
d/dx(ln(g(x)))=1/(g(x))*g'(x)
The given function is
f'(x)=1/(x-x^2)*d/dx(x-x^2)
f'(x)=1/(x-x^2)*(1-2x)
f'(x)=(1-2x)/(x-x^2)
To find the second derivative, use the quotient rule, which states that
d/dx((f(x))/(g(x)))=(g(x)f'(x)-f(x)g'(x))/(g(x))^2
Applying this to
f''(x)=((x-x^2)d/dx(1-2x)-(1-2x)d/dx(x-x^2))/(x-x^2)^2
Simplify through the power rule.
f''(x)=((x-x^2)(-2)-(1-2x)(1-2x))/(x-x^2)^2
f''(x)=(-2x+2x^2-(1-4x+4x^2))/(x-x^2)^2
f''(x)=(-2x^2+2x-1)/(x-x^2)