Nth Derivative of xⁿyⁿ by Leibniz Theorem | Solved Example

Nth Derivative of xⁿyⁿ Using Leibniz Theorem

In this problem, we evaluate the n^th derivative of xⁿyⁿ using the Leibniz Theorem. This method is commonly applied to higher-order derivatives involving products of functions.

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Problem

Find the n^th derivative of y = xⁿ · yⁿ using Leibniz theorem.

Solution

Leibniz Theorem

dⁿ/dxⁿ(uv) = Σ C(n, r) · d^ru/dx^r · d^n−rv/dx^n−r,
where r = 0 to n

Step 1: Choose Functions

u = xⁿ
v = yⁿ

Step 2: Derivatives of Each Function

Derivative of u = xⁿ:

d^r/dx^r(xⁿ) = n(n−1)(n−2)…(n−r+1)x^n−r

Derivative of v = yⁿ:

d^n−r/dx^n−r(yⁿ) = n(n−1)(n−2)…(r+1)y^r

Suggested Calculus Problems

• Nth Derivative of a^xb^x sinx
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• Nth Derivative of e^2x sin(3x)
• Nth Derivative of xⁿyⁿ

Step 3: Apply Leibniz Theorem

dⁿ/dxⁿ(xⁿyⁿ) = Σ C(n, r) · [n(n−1)…(n−r+1)x^n−r] · [n(n−1)…(r+1)y^r]

This is the required general expression.