Derivative of cos(u)

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 $f^{\prime }$ of the function $f(x)=\cos (u(x))$ is:

$$f^{\prime }(x)=-\sin (u(x))\cdot u^{\prime }(x)$$

ie

$$f^{\prime }=-\sin (u)\cdot u^{\prime }$$

where $u(x)$ is a differentiable function of $x$ and $u^{\prime }(x)$ 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:

$$\begin{array}{rl}(\cos (u(x)){)}^{\prime }& =-\sin (u(x))\cdot (u(x){)}^{\prime }\\ & =-\sin (u(x))\cdot u^{\prime }(x)\end{array}$$

This concludes the demonstration.