This equation shows the definition of a derivative. A derivative models the "instantaneous rate of change" of some function, with respect to a variable of that function.
| \( x \) | This is a symbol for any generic variable. It can hold any value, whether that be an integer or a real number, or a complex number, or a matrix etc. |
| \( f'(x) \) | This is a general symbol for the derivative of a general function, \(\htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x})\), in Legrange's notation. |
| \( h \) | This is the symbol for a step size in the input of a function. It is used when calculating a derivative by definition. |
| \( f \) | This is the symbol for a function. It is commonly used in algebra, and (multivariate) calculus. |
| \( \lim \) | This is the symbol for a limit in calculus. It's a function that models an output, as an input approaches a certain value. |
Consider some function \(\htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x})\). We are trying to find the derivative at some point \(\htmlClass{sdt-0000000003}{x} = \htmlClass{sdt-0000000121}{a}\), \( \htmlClass{sdt-0000000065}{f'(x)} \). We can imagine that the "instantaneous rate of change" that we are after will be the gradient (\( \htmlClass{sdt-0000000131}{X} \)lineGradient) of the tangent to \(\htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x})\) at \( \htmlClass{sdt-0000000121}{a} \). This can be seen in the following figure.
It will be quite challenging to get the gradient of this tangent, especially to find some general expression for it. What we can do much more easily, however, is to find the gradient of some segment of the graph between our point, \(\htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000121}{a})\), and some other point, \(\htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000121}{a} + \htmlClass{sdt-0000000067}{h})\). This can be seen in the following figure:
To get the gradient of this segment, we can use the formula for the gradient (\( \htmlClass{sdt-0000000131}{X} \)lineGradient) of a straight line:
\[\htmlClass{sdt-0000000138}{m} = \frac{\Delta \htmlClass{sdt-0000000017}{y}}{\Delta \htmlClass{sdt-0000000003}{x}}\]
\(\Delta \htmlClass{sdt-0000000017}{y}\) will be: \(\htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000121}{a} + \htmlClass{sdt-0000000067}{h}) - \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000121}{a})\)
\(\Delta \htmlClass{sdt-0000000003}{x}\) will be: \((\htmlClass{sdt-0000000121}{a} + \htmlClass{sdt-0000000067}{h}) - \htmlClass{sdt-0000000121}{a} = \htmlClass{sdt-0000000067}{h}\)
From here it follows that the gradient is:
\[ \htmlClass{sdt-0000000138}{m} = \frac{\htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000121}{a} + \htmlClass{sdt-0000000067}{h}) - \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000121}{a})}{\htmlClass{sdt-0000000067}{h}} \]
A generic expression for the gradient of the segment. To go from this equation to an equation for the gradient of the tangent, we would need the value of \( \htmlClass{sdt-0000000067}{h} \) to equal \(0\). Unfortunately, we can't just plug \(\htmlClass{sdt-0000000067}{h} = 0\) into the equation, because the denominator is \(\htmlClass{sdt-0000000067}{h}\) and dividing by \(0\) is undefined. However, what we can use are limits (\( \htmlClass{sdt-0000000101}{\lim} \)):
\(\lim\) is the symbol for a limit in calculus. It's a function that models an output, as an input approaches a certain value. For example, \( \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}) = \lim_{\htmlClass{sdt-0000000003}{x} \to 0^{+}} \frac{2}{\htmlClass{sdt-0000000003}{x}} = \infty\) would read:
'The function f, with respect to x, is equal to the limit as x approaches 0 from the positive direction of 2 divided by x is equal to infinity'
It can be a limit from the positive direction, meaning that if you have a limit of a function \(\htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x})\) on the \(xy\)-plane it is right to left. This is expressed as \(\lim_{\htmlClass{sdt-0000000003}{x} \to \htmlClass{sdt-0000000017}{y}^{+}}\). It can also be from the negative direction, meaning it is going left to right. This is expressed as: \(\lim_{\htmlClass{sdt-0000000003}{x} \to \htmlClass{sdt-0000000017}{y}^{-}}\).
It can also be coming from both directions. This value only exists if the limit from the positive direction has the same value as the limit from the negative direction. This is more simply expressed as: \(\lim_{\htmlClass{sdt-0000000003}{x} \to \htmlClass{sdt-0000000017}{y}}\).
Using limits, we can instead of asking what happens when \( \htmlClass{sdt-0000000067}{h} \) is \(0\), instead ask what happens when \( \htmlClass{sdt-0000000067}{h} \) gets closer and closer to \(0\), inevitably being infinitely close to zero. We can therefore take the \(\htmlClass{sdt-0000000101}{\lim}_{\htmlClass{sdt-0000000067}{h} \to 0}\) of our equation for the gradient of a segment to get:
\[ \htmlClass{sdt-0000000065}{f'(x)} = \htmlClass{sdt-0000000101}{\lim}_{\htmlClass{sdt-0000000067}{h} \to 0} \frac{\htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x} + \htmlClass{sdt-0000000067}{h}) - \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x})}{\htmlClass{sdt-0000000067}{h}} \]
as required.
Coming soon...