# Calculus of vectors¶

## Differentiation¶

### Vector-valued functions of scalars¶

A vector-valued function of a scalar (a real mouthful to say) is a function which outputs a vector value and is a function of a scalar variable. Given a function, $\mathbf{a}(x)$, the derivative of $\mathbf{a}(x)$ is similar to the derivative of a scalar valued function of a scalar. $$\frac{d\mathbf{a}}{dx} = \lim_{\Delta x\to 0} \frac{\mathbf{a}(x + \Delta x) - \mathbf{a}(x)}{\Delta x}$$ The rules for calculating the derivatives of such functions draw parallels with the rules for calculating the derivatives of scalar-valued functions. $$\frac{d}{dx}(h \cdot g) = \frac{dh}{dx} \cdot g + h \cdot \frac{dg}{dx}$$