What is the derivative of f(x)=4 cos ( 5x- 2). Please explain in detail.

For a function f(x) = g(h(x)), express h(x) as
y.

Then f(x) = g(y), f’(x) = [d {g(y)}/
dy]*(dy/dx).

Here we have to find the derivative of f(x)= 4
cos (5x-2).

Let y=5x-2, this gives f(x)= 4 cos
y

f’(x)= [d (4 cos
y)/dy]*[d(5x-2)/dx]

We also know that the derivative of cos
x= -sin x.

=>   [d (4 cos y)/dy]= -4 sin
y

[d(5x-2)/dx]= 5

Therefore
f’(x)= [d (4 cos y)/dy]*[d(5x-2)/dx]

= (-4 sin
y)*5

=-4*sin (5x-2)*5

=-20 sin
(5x-2)

Therefore the derivative of 4 cos
(5x-2) is -20 sin (5x-2)