Definition of the Derivative
- 01
What is a Derivative?
A derivative is the instantaneous rate of change of a function. This lesson introduces it as the slope of the tangent.
- 02
Rules for Differentiation
Rules for differentiation cover the power rule, the product rule, and the chain rule. This lesson states each one.
- 03
Rules for Calculating the Derivative of a Function
Rules for calculating the derivative of a function cover the power rule, the sum rule, and constant multiples.
- 04
Derivative by Definition 1
Derivative by definition means the limit of the difference quotient. This lesson is the first worked example.
- 05
Derivative by Definition 2
Derivative by definition means taking the limit of the difference quotient. This lesson is a second worked example.
- 06
Derivatives - Excercise 1
This derivatives exercise is the first practice set. It uses the definition and the basic rules on simple functions.
- 07
Derivative – Exercise 2
This derivative exercise is a second worked example. It applies the definition through the difference quotient.
- 08
Derivative – Exercise 3
This derivative exercise is a third set of practice. It applies the definition and the basic rules to new functions.
INTRODUCTION TO DIFFERENTIAL CALCULUS
In mathematical analysis, one of the key concepts is that which describes the change of a function with respect to the change in its argument. This concept leads to the formal definition of the derivative, which is the foundation for many further mathematical constructions and theories. The central idea is that the derivative measures the rate of change of a function at a given point – that is, how quickly and in which direction the function is changing there.
FORMAL DEFINITION
Let 'f' be any real function and x₀ be some point in its domain. The derivative of the function 'f' at the point x₀ is denoted as f′(x₀) and is defined as the limit of the difference quotient, if this limit exists:
f′(x₀) = lim (as h → 0) [f(x₀ + h) – f(x₀)] / h.
In this expression, 'h' represents a small change in the independent variable x, while the numerator shows the change in the function's value. If this limit exists, we say that the function is differentiable at that point.
GEOMETRIC MEANING OF THE DERIVATIVE
Geometrically, the derivative represents the slope of the tangent line to the graph of the function at the point x₀. If we imagine the graph of the function as a smooth curve, then the tangent at a given point indicates the direction in which the curve is heading. The larger the derivative, the steeper this slope.
EXAMPLE: DERIVATIVE OF THE FUNCTION F(X) = X²
Let's use the definition:
f′(x) = lim (as h → 0) [(x + h)² – x²] / h = lim (as h → 0) [x² + 2xh + h² – x²] / h = lim (as h → 0) [2xh + h²] / h = lim (as h → 0) (2x + h) = 2x.
Therefore, the derivative of the function x² is 2x. This means that the values of this function at point x are changing at a rate of 2x.
CONCLUSION
The derivative is a mathematical tool for describing the rate of change of a function. Using the limit of the difference quotient, we arrive at a precise definition that has both a numerical and a geometric meaning. By understanding basic examples, such as the quadratic function, we can more easily build knowledge about more complex functions.