can you please explain me leibnitz theorem ? i mean thru examples to get a better understanding .

Leibniz's theorem is a useful rule for finding the nth derivative of the product of two functions. It’s based on the product rule for differentiation but extended to higher-order derivatives.

Leibniz Theorem Statement
If
𝑢
(
𝑥
)
u(x) and
𝑣
(
𝑥
)
v(x) are two differentiable functions, then the nth derivative of their product is given by:

dⁿ/dxⁿ [u(x) v(x)] = Σ (nCk) * (dⁿ⁻ᵏ u/dxⁿ⁻ᵏ) * (dᵏ v/dxᵏ)

Where:

Σ represents the summation from k = 0 to n
nCk is the binomial coefficient (combinatorial term)
dⁿ⁻ᵏ u/dxⁿ⁻ᵏ and dᵏ v/dxᵏ are derivatives of respective orders
Example 1: Find the 3rd derivative of y = x² e^x
Let:

u(x) = x²
v(x) = e^x

Now apply Leibniz's theorem:

d³/dx³ [x² e^x] = Σ (3Ck) * d³⁻ᵏ(x²)/dx³⁻ᵏ * dᵏ(e^x)/dxᵏ

Step 1: Compute the derivatives
d⁰(x²)/dx⁰ = x²
d¹(x²)/dx¹ = 2x
d²(x²)/dx² = 2
d³(x²)/dx³ = 0

d⁰(e^x)/dx⁰ = e^x
d¹(e^x)/dx¹ = e^x
d²(e^x)/dx² = e^x
d³(e^x)/dx³ = e^x

Step 2: Combine terms using Leibniz's formula
d³/dx³ [x² e^x] = (3C0)(x²)(e^x) + (3C1)(2x)(e^x) + (3C2)(2)(e^x) + (3C3)(0)(e^x)

Step 3: Calculate each term
= (1)(x²)(e^x) + (3)(2x)(e^x) + (3)(2)(e^x) + 0
= x² e^x + 6x e^x + 6e^x

Step 4: Final Answer
= e^x(x² + 6x + 6)

Example 2: Find the 4th derivative of y = sin(x) * cos(x)
Let:

u(x) = sin(x)
v(x) = cos(x)

Now apply Leibniz's theorem:

d⁴/dx⁴ [sin(x) cos(x)] = Σ (4Ck) * d⁴⁻ᵏ(sin(x))/dx⁴⁻ᵏ * dᵏ(cos(x))/dxᵏ

Compute the derivatives:

d⁰(sin(x))/dx⁰ = sin(x)
d¹(sin(x))/dx¹ = cos(x)
d²(sin(x))/dx² = -sin(x)
d³(sin(x))/dx³ = -cos(x)
d⁴(sin(x))/dx⁴ = sin(x)

d⁰(cos(x))/dx⁰ = cos(x)
d¹(cos(x))/dx¹ = -sin(x)
d²(cos(x))/dx² = -cos(x)
d³(cos(x))/dx³ = sin(x)
d⁴(cos(x))/dx⁴ = cos(x)

Now substitute back:

= (4C0)(sin(x))(cos(x)) + (4C1)(cos(x))(-sin(x)) + (4C2)(-sin(x))(-cos(x)) + (4C3)(-cos(x))(sin(x)) + (4C4)(sin(x))(cos(x))
= (1)(sin(x))(cos(x)) + (4)(cos(x))(-sin(x)) + (6)(-sin(x))(-cos(x)) + (4)(-cos(x))(sin(x)) + (1)(sin(x))(cos(x))

Combining like terms:
= sin(x) cos(x) - 4 sin(x) cos(x) + 6 sin(x) cos(x) - 4 sin(x) cos(x) + sin(x) cos(x)
= 0

Key Points to Remember
Leibniz’s theorem simplifies differentiation of product functions for higher-order derivatives.
Carefully compute each derivative term and apply the binomial coefficients.
This method is especially useful for polynomial, exponential, and trigonometric functions.
If you’d like another example or further explanation, feel free to ask!