• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

The Gradient Function

Extracts from this document...


Jessica Mead        The Gradient Function        

The Gradient Function


In this project, I intend to investigate the Gradient Function and then prove how the formula that I get works.  The Gradient Function is an important part of calculus and mathematics as a whole.  It allows us to calculate accurately the gradient of any point on any graph without using the lengthy procedure of drawing a graph and then a tangent which is still extremely inaccurate – the error can be 20 units.  

To investigate the Gradient Function, there are two aspects of mathematics that must be made clear from the beginning: what is a gradient? and what is a tangent?

A gradient can be described as the steepness of a curve or line and is found by the formula:

Gradient of a         change in y

straight line           change in x

This is shown in the graph on the left.

A tangent  is defined as a straight line that just ‘touches’ the circumference of a circle (as shown in the picture on the right).  It is often used in mathematics to find an estimate of the gradient of a curve as it is a straight line, using the formula above.

Investigating a simple curved graph

I decided to start with the simplest curved graph, x2 to begin my investigation.













...read more.


Therefore the gradient of the graph y = x2 is 2x + d.

At first, I thought that this must be wrong because it was not simply 2x.  However, I realised that d was so small that in the context of finding the gradient, it was almost negligible.  Hence, to calculate the gradient normally it is simply Gradient  = 2x.

*        *        *

I was not able to find a formula for the graph x3, so I decided to use the above method to find out what it was.  

 Gradient = (x + d)3 –x3

      x + d – x

I expanded the brackets using Pascal’s Triangle and the Binomial Theorem which will be mentioned after I have found the formula for y = x3.

Gradient = x3 + 3x2d + 3xd2 + d3 – x3

                     x + d – x

As with the previous example I can cancel out the x3s and the xs. Therefore:

Gradient = 3x2d + 3xd2 + d3


All of the expressions on the top have d in, so I can factorise the top half of the equation:

Gradient = d(3x2 + 3xd + d2)


Thus, having cancelled out the ds (as shown):

the formula for the gradient of y = x3 is 3x2 + 3xd + d2

...read more.


δ    cos 0.5 = cos (0.5 +0.00001) – cos 0.5

δx                        0.00001

               = cos(0.50001) – cos 0.5


= - 0.000004794


= -0.4794298


Having investigated the gradient function for graphs of the type xn, I have come to the conclusion that the formula nx(n-1) is true for all types that I have investigated except for the trigonometrical graphs.  I have proved that this is true graphically, numerically and algebraically.  I have looked at tables of values, found formulas for each type of graph and then proved that these formulas are true, using algebra.  

I have furthered my investigation by looking at the gradient of trigonometrical graphs and shown that the theory that I found in a textbook is true.  Naturally, this investigation could be expanded further and further, looking at all types of curves such as negative curves and those to the power of different numbers.  

I have completed the objectives that I set myself at the beginning of the investiagtion.  I am now able to calculate any gradient accurately of any point on almost any graph without drawing a tangent.  I have also completed the objective of proving my formulas and generalising the rule.  Thus to find the gradient the formula is


...read more.

This student written piece of work is one of many that can be found in our GCSE Gradient Function section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Gradient Function essays

  1. Peer reviewed

    The Gradient Function Coursework

    5 star(s)

    This formula, which I found out to be m=nax^n-1 is called the derivative or, more helpfully, the gradient function. The process of finding the derivative is what is known as differentiation and belongs under the topic of calculus. But let us go back to the chord method.

  2. Analysing Triangle Vertices and Bisectors

    Since the point of intersection has co-ordinates that satisfy both equations, it is possible to substitute one of the original equations into the other to show the co-ordinates of C. Perpendicular bisector of AB is x = 7 Perpendicular bisector of OA is 3y = 10 - x Substituting x

  1. I have been given the equation y = axn to investigate the gradient function ...

    table is 3x� and when we use it here I get 3(3�) which is = 27 When y =2 x3 and the tangents and their respective gradients are as follows: Tangent at x = 2 3 4 Gradient function 24 54 96 Again I am going to establish a relationship between the tangent and the gradient.

  2. I am going to investigate the gradients of different curves and try to work ...

    y = 3x2 I am investigating the changes in gradient for the curve y = 3x2. To plot the curve, I will use the table of values given below. x 0 1 2 3 4 5 6 y 0 3 12 27 48 75 108 I will be working out

  1. The Gradient Function Investigation

    h = 2x + h (cancel h) As the h value then tends to (gets closer to) 0, so the gradient function for the graph y = x� tends to 2x. This rule can then be applied to any point on the curve and will give the gradient at the point precisely.

  2. Gradient Function

    and (-3, 9). From my results I can see that the closer I get to the fixed points, the gradient gets closer to double the value of X co-ordinate. Using the information I can conclude that the gradient on any point on that curve will be two times the X co-ordinate.

  1. The Gradient Function.

    dg/dc This should then give you an accurate gradient, it tends to be more accurate than the other method although if you were to draw the tangent and graph perfectly you should get the exact answer. One of the good things about this method is that it isn't necessary to sketch the magnified area instead there is another method.

  2. The Gradient Function

    I have drawn the curve of y = x2 in the colour of blue. I have decided to investigate two different tangents of the curve, to see if I can spot any patterns. The tangent at x = -1 has been marked in green, and the tangent at x = 1.5 has been marked in red.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work