Nth Derivative of x/((x−a)(x−b)(x−c)) Using Partial Fractions

Nth Derivative of x / (x − a)(x − b)(x − c)

To find the required n^th derivative of the given function, we first decompose it using the Partial Fraction Method. After decomposition, we apply the successive differentiation method.

Step 1: Partial Fraction Decomposition

Let

x / (x − a)(x − b)(x − c) = A / (x − a) + B / (x − b) + C / (x − c)

x = A(x − b)(x − c) + B(x − a)(x − c) + C(x − a)(x − b)

Step 2: Finding Constants A, B and C

Putting x = a:
a = A(a − b)(a − c)
A = a / (a − b)(a − c)

Putting x = b:
b = B(b − a)(b − c)
B = b / (b − a)(b − c)

Putting x = c:
c = C(c − a)(c − b)
C = c / (c − a)(c − b)

Step 3: Decomposed Form

x / (x − a)(x − b)(x − c) =

a / (a − b)(a − c) · 1/(x − a)
+ b / (b − a)(b − c) · 1/(x − b)
+ c / (c − a)(c − b) · 1/(x − c)

Recommended Calculus Problems

• Integration by Partial Fraction
• Integration by Parts – Solved Example
• Integration of e^ax sin(bx)

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

Step 4: N^th Derivative

We know that:

dⁿ/dxⁿ [1 / (x − a)] = (−1)ⁿ · n! / (x − a)ⁿ⁺¹

Applying this to each term:

dⁿ/dxⁿ [ x / (x − a)(x − b)(x − c) ] =

(−1)ⁿ n! · [
a / (a − b)(a − c)(x − a)ⁿ⁺¹
+ b / (b − a)(b − c)(x − b)ⁿ⁺¹
+ c / (c − a)(c − b)(x − c)ⁿ⁺¹ ]

This is the required n^th derivative.