If f ( x ) = sin 3 x, find f ' (3)

Answer 1

If f ( x ) = sin 3 x, find f ' (3) = ?? Radian

Detailed Solution:

f ^' ( x ) = cos ( 3x ) * ( 3 ) <== Chain Rule

f ^' ( x ) = 3 cos ( 3x )

f ^' ( 3 ) = 3 cos ( 3(3) )

f ^' ( 3 ) = 3 cos ( 9 ) <== Exact Solution in Radian

f ^' ( 3 ) = 3 cos ( 9 ) = –2.733390786 <== Approximation in Radian

Answer 2

Our main goal here is to take the derivative of f(x) with respect to x (and then plug in 3).

Let's look at another example. Let f(x) = (10x)^2

Note there are a couple ways we can find f'(x).

(1) One way to solve is to simplify f(x):

f(x) = (10x)^2 = (10x) * (10x) = 100x^2 -> f'(x) = 200x

(2) The other way to solve is to use a powerful concept called The Chain Rule (please look this up if you are not familiar)

Let y = 10x -> f(y) = y^2 -> so we know f'(y) = 2y = 2(10x) = 20x. BUT WAIT our answer does not agree with the answer we got in method (1). So why do my answers not agree?

Figuring out what I did wrong in (2) might help put you on the right track for solving your problem. Feel free to ask any follow up questions :)