Cartesian Coordinates Show

Gradient of a straight line Show

Equation of a straight line Show



  • 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: f(x) = ...
      • Then it's derivative is: f'(x) = ...
      • This is pronounced "f prime of x".
      • It is called "Lagrange's notation".
    • if a function is given in the form: y = ...
      • Then it's derivative is: dy/dx = ...
      • This is pronounced "dy over dx" or "dy by dx".
      • It is called "Leibniz's notation".
      • It is also sometimes written as: d/dx y = ...

Derivative of a straight line

  • We know that the equation of a straight line is: y = mx + b
  • 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: dy/dx = m

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) = x2 f(x) = x^2
  • If we examine the gradients at various points on the line: f(x) = x^2 with tangents at (-2, 4) and (1, 1)
    • 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, x2).
  • 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: x + \Delta
      • Our second point is then: (x, (x + delta)^2)
  • Our equation for the gradient is: equation for a straight line
  • Which simplifies to: simplified equation for a straight line And then: even more simplified equation for a straight line
  • As delta approaches 0, the equation becomes: dydx = 2x