In order to be able to deduce the derivative of the natural logarithm we resort to using implicit differentiation.
Let x= ey(x)
Differentiating both sides gives
dx/dx = d ey(x)/dx
Solving for dy(x)/dx one obtains
dy(x)/dx = 1/ey(x) = 1/x .
The product rule is given by
d f(x) g(x)/dx = df(x)/dxg(x) + f(x) dg(x)/dx
A beautiful proof for this theorem is given by G.W. Leibniz:
(u + du) *(v+dv) = u*v + u*dv + v* du + du*dv.
The last term is the product of two infinitesimals and can therefore be neglected. The differential of the product of the two functions u(x)*v(x) is thus equal to
u(x)*dv + v(x)*du.
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The rule for differentiation of the quotient of two functions can be deduced from the product rule and is given by
df(x)/g(x) =( df(x)/dx g(x) – f(x) dg(x))/dx)/(g(x))2.