find the derivative of f(x)= (3xsin4x)+ ((3/4) cos4x)?

Question

i know the answer is 12xcos4x but how dont get the steps. Do we use the product rule or difference rule? How do we get the chain rule after that?

Answer 1

well u can differentiate both components separately:

for the first u will need the chain rule which will give
where a=3x and b=sin(4x)
(3x)*(4*cos(4x))+(3)*(sin4x)=
12xcos(4x)+3sin(4x)

the second part is one simple derivative giving
3*(-sin(4x))

summing up the two will then give 12x*cos(4x)

Answer 2

product rule for x sin4x

chain rule when dealing with sin4x and cos4x

also note that 3 and 3/4 are constants

f'(x) = 3[ x (cos4x) 4 + (sin4x) 1] + (3/4) (-sin4x) 4

= 12 x (cos4x) on simplification

Answer 3

f(x) = (3x).(sin (4x)) + (3/4).cos(4x)
f `(x) = 3.sin(4x) + 4 cos(4x).(3x) - 3.sin(4x)
f `(x) = (12x).(cos 4x)
Notes
(3x).(sin 4x) is a product and product rule is used to differentiate.
(3/4).cos 4x is a function of a function, the functions being cos x and 4x.These are differentiated in turn.