For the function Prove that f’(1) = 100 f’ (0).

Given,

Now, differentiating both sides w.r.t x –

∴ )

Using algebra of derivatives –

⇒

Use:

∴

⇒ f’(x) = x^{99 +} x⁹⁸ + ….. + x + 1

∴ f’(1) = 1⁹⁹ + 1⁹⁸ + … + 1 + 1 (sum of total 100 ones) = 100

∴ f’(1) = 100

As, f’(0) = 0 + 0 + ….. + 0 + 1 = 1

∴ we can write as

f’(1) = 100×1 = 100× f’(0)

Hence,

f’(1) = 100 f’(0) ….proved