Definition of the Derivative

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.