Find the derivative of xⁿ + axⁿ ⁻ ¹ + a² xⁿ ⁻ ² + ... + aⁿ ⁻ ¹x + aⁿ for some fixed real number a

Solution:

Let f (x) = xⁿ + ax^{n - 1} + a² x^{n - 2} + ... + a^{n - 1}x + aⁿ

d/dx f (x) = d/dx [xⁿ + ax^{n - 1} + a² x^{n - 2} + ... + a^{n - 1}x + aⁿ]

On using derivative formula d/dx (xⁿ) = nx^{n - 1}, we obtain

f' (x) = nx^{n - 1} + a (n - 1) x^{n - 2} + a² (n - 2) x^{n - 3} + .... + a^{n - 1} (1) + aⁿ (0)

Thus, f' (x) = nx^{n - 1} + a (n - 1) x^{n - 2} + a² (n - 2) x^{n - 3} + .... + a^{n - 1}

NCERT Solutions Class 11 Maths Chapter 13 Exercise 13.2 Question 6

Find the derivative of xⁿ + axⁿ ⁻ ¹ + a² xⁿ ⁻ ² + ... + aⁿ ⁻ ¹x + aⁿ for some fixed real number a

Summary:

The derivative of xⁿ + axⁿ ⁻ ¹ + a² xⁿ ⁻ ² + ... + aⁿ ⁻ ¹x + aⁿ for some fixed real number a is f' (x) = nx^{n - 1} + a (n - 1) x^{n - 2} + a² (n - 2) x^{n - 3} + .... + a^{n - 1}