Leibnitz theorem is an extension of the product rule in differentiation. It provides a formula to find the nth derivative of the product of two functions. This theorem is very useful for higher-order derivatives in calculus.
General Product Rule
For first derivative:
d/dx (uv) = u′v + uv′
Leibnitz Theorem for nth Derivative
If y = u(x) · v(x), then its nth derivative is given by:
Leibnitz Formula:
dⁿ/dxⁿ (uv) = Σ (nCk) · u⁽ⁿ⁻ᵏ⁾ · v⁽ᵏ⁾
This means:
y⁽ⁿ⁾ = nC0 u⁽ⁿ⁾v + nC1 u⁽ⁿ⁻¹⁾v′ + nC2 u⁽ⁿ⁻²⁾v″ + ... + nCn u v⁽ⁿ⁾
Explanation of Terms
- u⁽ʳ⁾ = rth derivative of u(x)
- v⁽ʳ⁾ = rth derivative of v(x)
- nCk = binomial coefficient = n! / [k!(n−k)!]
- The sum runs from k = 0 to k = n
Why Use Leibnitz Theorem?
It helps in:
- Finding higher-order derivatives quickly
- Handling functions that are products of simpler functions
- Solving problems in physics and engineering involving repeated differentiation
Special Cases
1. When one function is constant
If u = constant, then:
y⁽ⁿ⁾ = constant × v⁽ⁿ⁾
2. When one function becomes zero after some differentiation
If u is a polynomial of degree m, then u⁽ᵏ⁾ = 0 for k > m. Thus Leibnitz formula simplifies because terms beyond k = m vanish.
Examples (Conceptual Overview)
Example: Find the nth derivative of y = xⁿ eˣ
Take u = xⁿ and v = eˣ.
Repeated derivatives of xⁿ reduce degree each time; derivatives of eˣ remain eˣ.
Apply formula:
dⁿ/dxⁿ (xⁿ eˣ) = Σ (nCk) (xⁿ)⁽ⁿ⁻ᵏ⁾ (eˣ)⁽ᵏ⁾
Simplify using known derivatives of polynomial and exponential functions.
Example: Find y⁽ⁿ⁾ for y = sin x · eˣ
Take u = sin x, v = eˣ. Use Leibnitz formula and known cyclic derivatives of sin x.
Important Notes
- Leibnitz theorem is essentially the “generalised product rule.”
- Derivatives of polynomial functions eventually become zero.
- Derivatives of eˣ remain eˣ always.
- Apply binomial coefficients properly to avoid mistakes.
Practice Questions
- Find the nth derivative of y = xᵐ eˣ.
- Find the nth derivative of y = x⁵ sin x.
- Use Leibnitz theorem for y = (x² + 3x)(eˣ).
- Show that the nth derivative of y = eˣ cos x is periodic.
Detailed Notes:
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