Using the limit definition of the derivative, find the derivative of f(x)=sin(3x) at x=2π

First, we need to find the derivative f'(x) for all x, and then we will substitute x=2π into the function we find. To do this, we need to set up our limit:
f'(x) = lim [sin(3(x+h))-sin(3x)]/h as h-> 0
We then use the identity sin(A+B) = sinAcosB + sinBcosA to simplify the limit
f'(x) = lim [sin(3x)cos(3h) + sin(3h)cos(3x) - sin(3x)]/h as h-> 0

Because h tends to 0, we can assume that h is as small as we want it to be. This means we're allowed to use the small angle approximations for sin3h and cos3h to simplify the expression. Remember that sin3h ~= 3h as h is small, and cos3h ~= 1- (3h)2/2 = 1- 9h2/2. Hence:
f'(x) = lim [ (1-9h2/2)sin(3x) + 3hcos(3x) - sin(3x)]/h as h->0
f'(x) = lim [sin(3x) -(9h2/2)sin(3x) + 3hcos(3x) - sin(3x)]/h as h->0 (multiply out (1-9h2/2)sin(3x) into two terms)
f'(x) = lim [3hcos(3x) -(9h2/2)sin(3x)]/h as h->0 (cancel out the sin(3x) terms)
f'(x) = lim 3cos(3x) - (9h/2)sin(3x) as h->0 (divide through by h)
f'(x) = 3cos(3x) (because (9h/2)sin(3x) ->0 as h-> 0)


This gives us our general function. We can then plug in x=2π into f'(x) to get out final answer.So
f'(2π) = 3cos(32π)
f'(2π) = 3cos(6π)
f'(2π) = 3
1 = 3

So our final answer is f'(2π)=3

Answered by Maths tutor

3640 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

How to differentiate y=2x(x-2)^5 to find dy/dx?


The random variable J has a Poisson distribution with mean 4. Find P(J>2)


How can I determine the characteristics of a curve on an x-y set of axis (eg. points of intersection, stationary points, area under graph)?


Given f(x)=2x^3 - 2x^2 + 8x, find f'(x) and f"(x).


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

MyTutor is part of the IXL family of brands:

© 2025 by IXL Learning