## Calculus

### Differentiation Introduction

This allows us to measure the rate of change of one quantity with respect to another (slope of line if represented graphically). In coordinate geometry we use the slope of a line formula to work out the slope. If the line is curved then we use calculus.

To work out the slope at any point on the curve below we would ideally use the slope of a line formula but the denominator would be zero so we cannot do this. To circumvent this we use the theory of limits, which allows us to calculate the slope as the denominator (difference between x-values) gets as close as possible to zero. This results in the completed formula derived from first principles for the derivative (dy/dx or f'(x)) below).

Differentiation by rule allows for a quick method to find the derivative of polynomial functions.

#### L.C. Solution

Note:  Another Leaving certificate past paper question is answered in the video tutorial above.

Note: In the first step f(x) = the original function to be derived in expanded form. In the last step, the limit as h goes to zero, reduces all h terms to zero.