Calculus Derivatives

Question

Consider the function 𝑓:ℝ→ℝf:R→R, given by

𝑓(𝑥)=sin2𝑥^2/4𝑥 if  𝑥≠0 

f(x)=0 if x=0

 .

 Use the definition of the derivative as a limit to show that 𝑓 is differentiable at 𝑥=0 and find 𝑓′(0)

Answer 1

Hi! Great, it looks like you updated the function. Now f is continuous at x=0 so it may be differentiable! Let's check!

By definition of derivative,

f'(0) = lim_x->0 [f(x)-f(0)]/(x-0) = lim_x->0 (sin2𝑥^2/4𝑥 -0)/(x-0) = lim_x->0 sin2𝑥²/4𝑥²

= lim_x->0 1/2 (sin2𝑥²/2𝑥²)=

= ½ (1) = ½

f' is differentiable at x=0 and f'(0) = ½.

Please note that lim_x->0 (sin2𝑥²/2𝑥²)= 1 because lim_x->0 (sin𝑥/𝑥)= 1 (a well known limit).

I hope this helps! Please let me know if you have any questions! Good luck.