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) Show all steps.

Answer 1

f(x) = sin(2x) squared /4x by 1/2 angle formula f(x) = (1 -cos(4x))/8x. Now use McLaurin series for cos(x)

f(x) = (1/8x) (1 - sum(n=0 to inf){ ((-1)^(n-1)) )(4x)^(2n))/((2n)!) })

= (1/8x) sum(n=1 to inf) {((-1)^(n-1)) )(4x)^(2n))/((2n)!) }

= sum(n=1 to inf) {((-1)^(n-1)) )(4x)^(2n-1))/(2(2n)!) }

= x - (4x)^3/4) +-...

This function is differentiable for all x and is contiuous at x =0 with derivative = 1!