# What is the Derivative of Cos(x)?

## What is the Derivative of Cos(x)?

As in the calculation $$\frac{d}{dx} sin x$$,we begin with the definition of the derivative:

$$\frac{d}{dx} cos (x) = \lim_{d \to 0} \frac{(cos(x+d)- cos(x))}{d}$$

You may remember the following angle sum formula from high school:  $$cos(a+b) = cos(a)cos(b) – sin(a) sin(b)$$

$$\frac{d}{dx} cos (x) = \lim_{d \to 0} \frac{(cos(x)cos(d) – sin(x) sin(d) – cos(x))}{d}$$

$$\frac{d}{dx} cos (x) = \lim_{d \to 0} \frac{(cos(x)cos(d) – cos(x) – sin(x) sin(d))}{d}$$

$$\frac{d}{dx} cos (x) = \lim_{d \to 0} [ \frac{cos(x)cos(d) – cos(x)}{d} + \frac{-sin(x) sin(d)}{d} ]$$

$$\frac{d}{dx} cos (x) = \lim_{d \to 0} [ \frac{cos(x)(cos(d) – 1)}{d} + \frac{-sin(x) sin(d)}{d} ]$$

$$\frac{d}{dx} cos (x) = \lim_{d \to 0} [ \frac{cos(x)(cos(d) – 1)}{d} + (-sin(x)) (\frac{ sin(d)}{d} )]$$

$$\frac{d}{dx} cos (x) = \lim_{d \to 0} cos(x) (\frac{cos(d) – 1}{d} ) + \lim_{d \to 0} (-sin(x)) (\frac{ sin(d)}{d} )$$

Once again we use the following (unproven) facts:

$$\lim_{d \to 0} \frac{cos(d) – 1}{d} = 0$$

$$\lim_{d \to 0} \frac{ sin(d)}{d} = 1$$

$$\frac{d}{dx} cos (x) = \lim_{d \to 0} cos(x) (\frac{cos(d) – 1}{d} ) + \lim_{d \to 0} (-sin(x)) (\frac{ sin(d)}{d} )$$

$$\frac{d}{dx} cos (x) = cos(x) \times 0 + (-sin(x)) \times 1$$

$$\frac{d}{dx} cos (x) = 0 + (-sin(x))$$

$$\frac{d}{dx} cos (x) = -sin(x)$$