Introduction to Calculus

Differentiation and The Derivative

Introduction

Calculus is a very important branch of mathematics. It is a form of mathematics applied to continuous graphs (graphs without gaps). Calculus has two aspects:

  • Differentiation (finding derivatives of functions)
  • Integration (finding indefinite integrals or evaluating definite integrals)

Calculus was invented by European mathematicians, Isaac Newton and Gottfried Leibnitz.

This article introduces Differentiation.

The derivative allows us to calculate the slope or tangent of a graph at any point, P. The process by which a derivative is found is called Differentiation.

The graph below is a simple parabola whose equation is y = x2.

The derivative is given the symbol

\frac{dy}{dx}

This is pronounced d y by d x or dy dx.

The derivative is a function that gives the slope (tangent) of the graph at any point. The derivative measures the rate of change of y with respect to x. It describes in precise mathematical terms how y changes when x changes. This concept is very important in science.

It can be shown that if y = x2, then the derivative is given by

\frac{dy}{dx}=2x

So for this curve, when x = 1, the slope is 2; the slope at x = 3 is 6.

The derivative of y = x3 is found to be dy/dx = 3x2.

For y = x4, the derivative is dy/dx = 4x3.

Example 1: Find the slope of the curve y = x3 at the points x = -1 and x = 2 given that the derivative is 3x2.

The derivative of this curve is dy/dx = 3x2.

When x = -1, dy/dx = 3; when x = 2, dy/dx = 12.