How do you derive a trig function?

Question

This problem is giving me, well...problems:

f(x) = 3x csc(8 x)

How exactly would one derive this?

Answer 1

People seem to have trouble telling the difference between derive and differentiate. Differentiate is the actual correct word.
f(x) = 3x csc (8x)
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csc x = 1 / sin x
d/dx 1/sin x = -cosx / sin^2 x = (-cot x csc x)
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f'(x) = 3(csc (8x)) + (-8)(3x)(cot (8x) csc (8x))
f'(x) = 3(csc (8x)) - (24x)(cot (8x) csc (8x))

Answer 2

You derive it by simply writing it down.

It is the same as 3x/sin(8x)

It has the ideterminate form 0/0 when x = 0
The limit as x--> 0, however is 3/8.

Answer 3

i imagine, definite, you opt for to apply product rule. because even you modify it into 5xsinx + 5xcosx, they are nonetheless incorporates "product". assume we are saying: u = 5x du/dx = 5 v = (sin x + cos x) dv/dx = cos x - sin x Product rule : dy/dx = du/dx *v + u* dv/dx =5(sinx + cos x) + 5x(cos x - sin x) = .....