DIFFERENTIATION 01
Derivative from First Principles
The derivative is the limiting value of the average rate of change as the input interval shrinks toward zero.
The main idea
For a function
\[ y=f(x), \]the average rate of change over a finite input change \(\Delta x\) is
This uses two different points on the curve.
1. The problem
To find the rate of change at one point, we need the second point to move closer to the first.
So we let
\[ \Delta x\to0. \]But we do not set \(\Delta x=0\) inside the fraction, because that would divide by zero.
2. The definition
The derivative of \(f\) at \(x\) is defined by
provided this limit exists.
3. What this means geometrically
For \(\Delta x\neq0\), the fraction
\[ \frac{f(x+\Delta x)-f(x)} {\Delta x} \]is the gradient of a secant line.
As
\[ \Delta x\to0, \]the second point approaches the first, and the secant line approaches a tangent line.
The derivative is this limiting gradient.
4. Diagram
5. Example: \(f(x)=x^2\)
Let
\[ f(x)=x^2. \]Using the definition,
\[ f'(x) = \lim_{\Delta x\to0} \frac{(x+\Delta x)^2-x^2} {\Delta x}. \]Expand:
\[ (x+\Delta x)^2 = x^2+2x\Delta x+(\Delta x)^2. \]Therefore,
\[ f'(x) = \lim_{\Delta x\to0} \frac{2x\Delta x+(\Delta x)^2} {\Delta x}. \]For \(\Delta x\neq0\),
\[ f'(x) = \lim_{\Delta x\to0} \left(2x+\Delta x\right). \]Now take the limit:
6. What \(f'(x)=2x\) means
The derivative is itself a function.
For
\[ f(x)=x^2, \]we found
\[ f'(x)=2x. \]So the slope depends on the point.
At \(x=1\),
\[ f'(1)=2. \]At \(x=3\),
\[ f'(3)=6. \]The curve becomes steeper as \(x\) increases.
7. Derivative at a particular point
If we want the derivative at \(x=a\), we can write
Here \(h\) plays the same role as \(\Delta x\).
8. Why \(h\) is often used
The letters
\[ h \qquad\text{and}\qquad \Delta x \]are both commonly used for the finite input change in the difference quotient.
So
\[ \lim_{h\to0} \frac{f(x+h)-f(x)}{h} \]and
\[ \lim_{\Delta x\to0} \frac{f(x+\Delta x)-f(x)} {\Delta x} \]express the same definition.
9. The derivative as a rate of change
The derivative is not only a slope. It also gives an instantaneous rate of change.
If \(s(t)\) is position, then
\[ s'(t) \]is instantaneous velocity.
If \(P(t)\) is population, then
\[ P'(t) \]is the instantaneous population growth rate.
10. The central idea
This is the conceptual foundation of differentiation.