A Brief Introduction to Differential Calculus

Recall that the slope is defined as the change in Y divided by the change in X

Consider the straight line below: Y X

Let’s consider a line drawn between two points on a curve. We start at (X 0,Y 0 ). Then we change our X value slightly to X 0 + ΔX and our Y value to the corresponding value, so we are now at (X 0 + ΔX, Y 0 + ΔY). Y Y 0 Y 0 + ΔY X 0 X 0 + ΔX X

Then ΔY/ ΔX is the slope of the line connecting the two points. Y Y 0 Y 0 + ΔY X 0 X 0 + ΔX X

If we shrink ΔX a bit, our picture looks like this: X 0 X 0 + ΔX X Y Y 0 Y 0 + ΔY

If we make ΔX infinitesimally small, then X 0 + ΔX is virtually identical to X 0, Y 0 + ΔY is virtually identical to Y 0, and we are looking at the line tangent to the curve. X 0 X Y Y 0

So the slope of a curve at a point is the slope of the line tangent to the curve at that point. Y X

Let’s calculate the derivative for the function, Y = 3X 2.

To calculate derivatives for similar functions of the form Y = aX n, we use the power function rule.

What is the derivative of a constant function Y = k (example: Y = 4)?

Notation

Example: Determine the derivative of Y = 6 + 2X 3 + 4X 5

There is a special product rule for determining the derivative of the product of functions. (We will not be examining that here.)

We have touched on a very small part of differential calculus. There is also a quotient rule for the derivative of the quotient of two functions. There is a chain rule for the derivative of a function of a function. There are rules for the derivatives of exponential functions, logarithmic functions, and trigonometric functions.

In this course, we will see how differential calculus is applied to Economics.