To differentiate cos(x^2-4x)cos(x2−4x), we have to apply the chain rule:
color(green)((f @ g)'(x) = f'(g(x)) * g'(x))
In other words:
1) Get the derivative of the outer function, and plug in the inner function...
2) Then multiply that by the derivative of the inner function.
In cos(x^2-4x), the outer function is cos x and the inner function is x^2 - 4x.
The derivative of cos x is -sin x, so we get:
d/dx cos(x^2-4x)
= -sin(x^2 - 4x) * d/dx (x^2 - 4x)
Next we solve d/dx (x^2 - 4x) using the power rule:
color(green)(d/dx x^n = nx^(n-1))
-sin(x^2 - 4x) * d/dx (x^2 - 4x)
= -sin(x^2 - 4x) * (2x - 4)
In summary:
d/dx cos(x^2-4x)
= -sin(x^2 - 4x) * d/dx (x^2 - 4x)
color(blue)(= -sin(x^2 - 4x) * (2x - 4))