A Brief Introduction to Differential Calculus

1. A Brief Introduction to Differential Calculus

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

3. Consider the straight line below: Y 20 6 5 12 X

4. Let’s consider a line drawn between two points on a curve. We start at (X0,Y0). Then we change our X value slightly to X0+ ΔX and our Y value to the corresponding value, so we are now at (X0+ ΔX, Y0 + ΔY). Y Y0 Y0 + ΔY X0 X0+ ΔX X

5. Then ΔY/ ΔX is the slope of the line connecting the two points. Y Y0 Y0 + ΔY X0 X0+ ΔX X

6. If we shrink ΔX a bit, our picture looks like this: Y Y0 Y0 + ΔY X0 X0+ ΔX X

7. If we make ΔX infinitesimally small, then X0+ ΔX is virtually identical to X0, Y0+ ΔY is virtually identical to Y0, and we are looking at the line tangent to the curve. Y Y0 X0 X

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

10. Let’s calculate the derivative for the function, Y = 3X2.

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

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

16. Notation

18. Example: Determine the derivative of Y = 6 + 2X3 + 4X5

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

20. 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.

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