Differentiate the following with respect to x:

a₀ xⁿ + a₁ x^{n – 1} + a₂ x^{n – 2} + ……. + a_{n – 1} x + a_n

Given,

f(x) = a₀ xⁿ + a₁ x^{n – 1} + a₂ x^{n – 2} + ……. + a_{n – 1} x + a_n

we need to find f’(x), so differentiating both sides with respect to x –

∴ (a₀ xⁿ + a₁ x^{n – 1} + a₂ x^{n – 2} + ……. + a_{n – 1} x + a_n)

Using algebra of derivatives –

⇒ f’(x) =

Use the formula:

∴ f’(x) = a₀ n x^{n – 1} + a₁ (n – 1) x^{n – 1 – 1} + a₂(n – 2) x^{n – 2 – 1} + ……. + a_{n – 1} + 0

⇒ f’(x) = a₀ n x^{n – 1} + a₁ (n – 1) x^{n – 2} + a₂(n – 2) x^{n – 3} + ……. + a_{n – 1}

∴ f’(x) = a₀ n x^{n – 1} + a₁ (n – 1) x^{n – 2} + a₂(n – 2) x^{n – 3} + ……. + a_{n – 1}