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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 up previous
Next: Standard results Up: Calculus - a brief Previous: What is calculus?
Alexander Frolkin 2001-03-13