In calculus, The Derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity.
1. Geometric Interpretation: Slope
Geometrically, the derivative of a function at a certain point is the slope of the tangent line to the graph of the function at that point.
Imagine drawing a line connecting two points on a curve. This is called a secant line. If you move those two points closer and closer together until they are virtually on top of each other, the secant line becomes the tangent line.
[Image of secant line becoming tangent line limit]The slope of this tangent line represents the Instantaneous Rate of Change.
2. The Formal Definition
We define the derivative using limits. This is known as the "Difference Quotient". It calculates the slope between two points, x and x+h, and then shrinks the distance h to zero.
[Image of difference quotient graph]f'(x) = lim (h→0) [ f(x + h) - f(x) ] / h
This formula is the bedrock of differential calculus. Every "shortcut" rule (like the Power Rule) is proven using this specific limit.
3. Notation
There are two common ways to write a derivative, named after the fathers of calculus:
- Lagrange's Notation: Uses a prime mark.
f'(x) or y'. This is useful for emphasizing the functional relationship. - Leibniz's Notation: Uses differentials.
dy/dx or d/dx [f(x)]. This is useful for remembering that a derivative is a ratio of tiny changes (change in y divided by change in x).
4. Differentiability vs. Continuity
A function must be continuous to be differentiable, but being continuous isn't enough. A function fails to have a derivative if it has:
- A sharp corner or cusp (like absolute value |x| at 0).
- A vertical tangent line (slope is infinite).
- A discontinuity (jump or hole).
At a sharp corner, you cannot draw a single unique tangent line; the "slope" is different depending on which side you approach from.