Table of Contents
Leibnitz Theorem For nth Derivative
If ƒ is a function and ƒ’, ƒ”, ƒ’’, ƒ’’’, ƒ’’’’ are its derivatives at a point, then
If ƒ is a differentiable function, ƒ’(x) exists at all points in the domain of ƒ, and ƒ is continuous at a point c, then
Theorem 1
The nth derivative of ƒ at c exists and is
ƒ’’’(c) = ƒ’(c)
Proof
The proof is by induction on n. The base case is n = 1, for which
ƒ’’’(c) = ƒ’(c)
The inductive step is to show that
ƒ’’’(c) = ƒ’’(c) ƒ’(c)
which is the same as
(ƒ’’’)’ = ƒ’’’’
But
(ƒ’’’)’ = ƒ’’’
Derivation of the Leibnitz Theorem/Formula
Let $f(x) = \frac{x^2}{2}$. The derivative of $f$ is $f'(x) = 2x$.
Now, let $g(x) = x^2$. The derivative of $g$ is $g'(x) = 2x$.
We want to find the derivative of $f(g(x))$.
This is equal to the derivative of $f$ multiplied by the derivative of $g$, or $f'(g(x)) = 2(g'(x))$.
Explain in Detail :
The figure below shows a simple bar chart that compares the number of passengers carried by bus and train in a certain city over the past five years.
The bar chart shows that the number of passengers carried by bus has been decreasing over the past five years, while the number of passengers carried by train has been increasing. In 2013, the number of passengers carried by bus was twice as high as the number of passengers carried by train, but by 2018, the number of passengers carried by train was twice as high as the number of passengers carried by bus.
