The derivative f’ of the function f(x)=cos(u(x)) is: f’(x) = -sin(u(x)) * u’(x) for any value of x.

Derivative of the Cosine Function of u

The derivative $f’$ of the function $f(x)=\cos(u(x))$ is:

\[f'(x) = -\sin(u(x)) \cdot u'(x)\]

ie

\[f'= -\sin(u) \cdot u'\]

where $u(x)$ is a differentiable function of $x$ and $u’(x)$ is its derivative.

Proof/Demonstration

The chain rule tells us that the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function. Applying this rule:

\[\begin{aligned} (\cos(u(x)))' &= -\sin(u(x)) \cdot (u(x))' \\ &= -\sin(u(x)) \cdot u'(x) \end{aligned}\]

This concludes the demonstration.