1. What a Derivative Really Means
A derivative tells how fast a function changes at a particular point. It measures the rate at which the output moves when the input changes slightly. In personal-notes style: the derivative is the ‘instantaneous speed’ of the function at any point.
It answers questions like “How steep is the curve here?” or “How quickly is this value rising or falling?”
2. Derivative as a Limit
The formal idea of a derivative comes from looking at the average rate of change over smaller and smaller intervals. As the interval shrinks, the average rate becomes the exact rate.
2.1. Limit Definition
f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
This expression is called the difference quotient. It captures the ratio of change in f to the change in x as the interval becomes extremely small.
3. Geometric Meaning: Slope of the Tangent
On a graph, the derivative at a point gives the slope of the tangent line at that point. The tangent line touches the curve at exactly one point (locally) and shows the direction the curve is heading.
So the derivative describes the steepness of the curve right at that location.
4. Difference Quotient View
The idea of taking the limit comes from the average rate of change:
4.1. Average Rate of Change
\frac{f(a+h) - f(a)}{h}
This is the slope of a secant line between two points of the graph.
4.2. Instantaneous Rate of Change
As h → 0, the secant line becomes the tangent line. The limit gives the exact slope at x = a.
5. When a Derivative Exists
A function is differentiable at a point if the limit in the definition exists as a finite number. Smooth curves satisfy this easily, but sharp edges or breaks do not.
5.1. Cases Where Derivative Fails
- Sharp corners (like |x| at x = 0)
- Cusps
- Vertical tangents
- Discontinuities
6. Basic Example of Finding a Derivative
Let’s compute the derivative of a simple function using the definition.
6.1. Example: f(x) = x²
f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}
= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}
= \lim_{h \to 0} \frac{2xh + h^2}{h}
= \lim_{h \to 0} (2x + h)
= 2x
So the derivative of x² is 2x.
7. Interpretation in Real Situations
Derivatives describe how quickly quantities change — like position changing over time (velocity), velocity changing over time (acceleration), or any measurable change. Think of the derivative as the ‘instant rate of change’ at an exact moment.