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Confusion can arise because we use numbers to represent the gradient of a line at a certain point (its rate of change or slope which is like its 'steepness') and other times we find an equation. The equation (the derivative) is used to find the gradient of a line at any point, also we can look at simple derivatives and see how quickly it changes or how it is changing.

The gradient is not the same as the steepness of a hill. When driving, it is more useful to see how much higher you are when you drive along the road. For a road that has a gradient of 1:1 means for every metre (or mile, km etc.) that you drive along the road, you will be a metre higher (or lower of course, if you are driving down it).

Although it may not seem like it, differentiation is the opposite of integration.

Notation

There are a few ways to write an equation down involving differentiation. You can say:

y = 2x

dy dx |
= 2 |

or

d dx |
2x = 2 |

The dx part is just a general symbol. You read it as "with respect to x" but you can use anything for example time, dt.

Let's look at this more closely.

dy

dx

This is not to be interpreted as one number divided by another but as a shorthand for an equation involving limits (more on this later). Leibniz (who introduced this notation) intended this to represent two infinitesimally small numbers (numbers that are greater than 0 but smaller than any number you can think of).

They can be separated though (a 'separation of variables' technique) to solve certain equations. This is because of the properties of limits. When defining differentiation with limits (explained below), the variables can be separated, so dy and dx can. Here is one property of limits which allows them to be separated:

lim | f(x) | = | lim f(x) | / | lim g(x) | |

x → p | g(x) | x → p | x → p |

Also you can also think of d / dx as an operator, something which takes a function and outputs a derivative.

Another notation that is used is:

if f(x) = sin x then f'(x) = cos x. The notation f' is read as "f prime" which may lead to some confusion as it just means the gradient or derivative.

You can say if f(x) = 2x then f'(5) = 10 meaning the gradient of the line at x = 5 is 10. This is the gradient at that one particular point.

If you just do the same thing with a curve, you get an approximation of the gradient at that point. If the triangle got smaller and smaller, the answer gets closer and closer to the true gradient (at that point).

If we look at y = x

If we want a point further on the line, we add these small numbers to y and to x:

y + Δy = (x + Δx)

This means:

Now I expand this equation so I can modify it:

y + Δy = (x + Δx)

What I want to do with this is to divide the difference in the y with the difference with the x or Δy /Δx which is a gradient.

I will subtract my original expression from the above to remove the y to leave Δy on the left of the equation:

y + Δy = x^{2} + 2x(Δx) +
(Δx)^{2}

minus

y = x^{2}

equals

Δy = 2x(Δx) + (Δx)^{2}

minus

y = x

equals

Δy = 2x(Δx) + (Δx)

Now I wish to make the Δy part in to Δy / Δx. I achieve this by dividing both sides by Δx:

Δy Δx |
= 2x | Δx Δx |
+ | (Δx)^{2}Δx |

Δy Δx |
= 2x + Δx |

This procedure cannot be used with everything though. Trigonometric expressions need substitution, logarithmic ones need a different strategy.

My graphical calculator

On my calculator, it can calculate the gradient at a point on a line. It does not use limits though.

One equation that can be used is to use a very small number for Δx (say, 10

f'(x) ≈ | f(x + Δx) - f(x) Δx |

f'(x) ≈ | f(x + Δx) - f(x - Δx) 2(Δx) |

Have you found an error or do you want to add more
information to these pages? You can contact me at the bottom of the home page. |