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Differential calculus

Differential calculus is concerned with rates of change. Geometrically, the rate of change of a function is defined as the ratio of the rise to the run, i.e. $\frac{\delta y}{\delta x}$ , in Cartesian coordinates where $\delta y$ and $\delta x$ are small increments in $x$ and $y$ . In calculus notation, the gradient of the curve is written

$\begin{displaymath} \frac{\mathrm{d}y}{\mathrm{d}x} \end{displaymath}$

(Note, that since the d's are operators rather than variables, they do not cancel.) Here, $\mathrm{d}x$ and $\mathrm{d}y$ are infintessimaly small changes in $x$ and $y$ respectively, called differentials.

$\includegraphics{definition.eps}$

The diagram above shows a curve with equation $y = f(x)$ . To find the gradient at P, i.e. $\frac{f(x+h) - f(x)}{h}$ , we need to let $h$ tend to 0. Hence, we can define the gradient of a function at a point by

$\begin{displaymath} \frac{\mathrm{d}y}{\mathrm{d}x} = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \end{displaymath}$

The gradient of a function is also known as its derivative. Commonly used notation includes $y'$ for the derivative with respect to $x$ and $\dot{y}$ for the derivative with respect to $t$ . (Note, that derivate and differential are not the same thing. The derivative of $y$ with respect to $x$ is $\frac{\mathrm{d}y}{\mathrm{d}x}$ , but the differential of $y$ is $\mathrm{d}y$ .)

Using the definition of the derivative above, we can deduce the gradient function of any function for which it is defined. For example, for the function $y = x^2$ ,

$\begin{eqnarray*} \frac{\mathrm{d}y}{\mathrm{d}x} & = & \lim_{h \to 0} \frac{(x+... ...c{2hx + h^2}{h} \\ & = & \lim_{h \to 0} 2x + h \\ & = & 2x \\ \end{eqnarray*}$

So to find the gradient of $y = x^2$ at the point $(3, 9)$ , we substitute the $x$ coordinate into the gradient function $\frac{\mathrm{d}y}{\mathrm{d}x}$ to obtain $\frac{\mathrm{d}y}{\mathrm{d}x} = 2 \cdot 3 = 6$ . The geometric significance of this is that at this point, for every unit change in $x$ , we need to increase $y$ by six units to remain on the tangent to the curve at this point.

Differentiation is a linear transformation. That is to say

$\begin{displaymath} \frac{\mathrm{d}}{\mathrm{d}x}(a f(x) + b g(x)) = a \frac{\mathrm{d}f}{\mathrm{d}x}(x) + b \frac{\mathrm{d}g}{\mathrm{d}x}(x) \end{displaymath}$

where a and b are constants.

Subsections

Next: Standard results Up: Calculus - a brief Previous: What is calculus?

Alexander Frolkin 2001-03-13