- #1

Karol

- 1,380

- 22

## Homework Statement

Use ##~\cos\alpha-\cos\beta~## to develop the formula for the derivative of cos(x) from the definition:

$$\frac{d(\cos x)}{dx}=\lim\frac{\cos(x+\Delta x)-\cos x}{\Delta x}$$

## Homework Equations

$$\cos\alpha-\cos\beta=(-2)\sin\left( \frac{\alpha+\beta}{2} \right)\cdot \sin\left( \frac{\alpha-\beta}{2} \right)$$

$$\lim\frac{\sin x}{x}=1$$

## The Attempt at a Solution

$$\lim\frac{\cos(x+\Delta x)-\cos x}{\Delta x}=...=(-2)\lim\frac{\sin\left( \frac{2x+\Delta x}{2} \right)}{\Delta x}\lim\frac{\sin\left( \frac{\Delta x}{2} \right)}{\Delta x}$$

Even if i could use ##~\lim\frac{\sin x}{X}=1~##, which would eliminate the second member, i can't deal with the first member.