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) = x99 + 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