Introduction
- A derivitive is a function that describes another function.
- More specifically:
- It describes the gradient of a function.
- A gradient is how much y changes in response to a change in x.
- Finding the derivitive is sometimes called differentiation.
- Notation:
- If a function is given in the form:
- Then it's derivative is:
- This is pronounced "f prime of x".
- It is called "Lagrange's notation".
- if a function is given in the form:
- Then it's derivative is:
- This is pronounced "dy over dx" or "dy by dx".
- It is called "Leibniz's notation".
- It is also sometimes written as:
- If a function is given in the form:
Derivative of a straight line
- We know that the equation of a straight line is:
- Where:
- m = the gradient of the slope.
- b = the point that the line intercepts the y axis.
- Straight lines are straight because their gradient doesn't change:
- For all points along the line, the gradient is the same. It is always m
- The derivative of a straight line is therefore:
Derivative of a curved line
- Curved lines are different to straight lines.
- Their gradient changes depending on where you measure it.
- for example: f(x) = x^{2}
- If we examine the gradients at various points on the line:
- At x = -2, the gradient is -4
- At x = 1, the gradient is 2
- To find the gradient of at a certain point we:
- Choose a point that is very close to the original.
- Draw a straight line between them.
- And then find the gradient of the line.
- For our first point we can choose (x, y), or more specifically (x, x^{2}).
- For our second point:
- We need a point very close to x
- Such as x + 0.1
- Or x + 0.01
- We use the greek letter delta to represent this value:
- Our second point is then:
- We need a point very close to x
- Our equation for the gradient is:
- Which simplifies to: And then:
- As delta approaches 0, the equation becomes: