Question 38 of 70intermediate🔧 ApplyShort Answer2 marks

Find the derivative of xn−anx−a\frac{x^n - a^n}{x - a} for some constant aa.

Correct Answer

The derivative is (n−1)xn−2+a(n−2)xn−3+a2(n−3)xn−4+…+an−2(n-1)x^{n-2} + a(n-2)x^{n-3} + a^2(n-3)x^{n-4} + \ldots + a^{n-2}.

Exercise: EXERCISE 12.2 | Q: 8 | (Page Page 32)
For More Understanding

Explanation

This question tests the student's understanding of algebraic simplification and differentiation of polynomials. The key insight from the textbook is the factorization identity for xn−anx^n - a^n, which allows simplification before differentiation. The derivative is then found by applying Theorem 6 (derivative of xnx^n is nxn−1nx^{n-1}) term by term.

Solution Steps

  1. Step 1: Use the factorization identity from the textbook: xn−an=(x−a)(xn−1+axn−2+a2xn−3+…+an−2x+an−1)x^n - a^n = (x - a)(x^{n-1} + ax^{n-2} + a^2x^{n-3} + \ldots + a^{n-2}x + a^{n-1})

  2. Step 2: Simplify the given expression: xn−anx−a=xn−1+axn−2+a2xn−3+…+an−2x+an−1\frac{x^n - a^n}{x - a} = x^{n-1} + ax^{n-2} + a^2x^{n-3} + \ldots + a^{n-2}x + a^{n-1}

  3. Step 3: Apply Theorem 6 (derivative of xnx^n is nxn−1nx^{n-1}) to each term. The derivative of xn−1x^{n-1} is (n−1)xn−2(n-1)x^{n-2}, derivative of axn−2ax^{n-2} is a(n−2)xn−3a(n-2)x^{n-3}, and so on.

  4. Step 4: The last term an−1a^{n-1} is a constant, so its derivative is 0. The final answer is: (n−1)xn−2+a(n−2)xn−3+a2(n−3)xn−4+…+an−2(n-1)x^{n-2} + a(n-2)x^{n-3} + a^2(n-3)x^{n-4} + \ldots + a^{n-2}

PreviousNext