Derivative of the natural logarithm by using the fundamental theorem of calculus.
To find the derivative of y = ln x. This solution below is not complete. It would be useful to use the Fundamental Theorem of Calculus here.
Integral(1, x) 1/t dt = F(x) - F(1), where F(x) is the antiderivative of 1/x (so F\'(x) = 1/x).
ln(x) = Integral(1,x) 1/t dt = F(x) - F(1)
Now take the derivative:
d/dx(ln(x)) = d/dx(F(x) - F(1)).
Continue to show that d/dx(ln(x)) = 1/x.