The increment method, which we use as it is more accurate than the tangent one. You zoom into the graph and take the x point and y point just after a whole figure. For instance with y-x2 , using point (1,1) we would do 1.0012 – 1 for the y value and 1.001 – 1 for the x value. So therefore –
Method = 0.002001 / 0.001 = 2.001, which we can say is roughly 2.
Y=X
Please see graph y=x
As the co-ordinates are the same, the distances between the parts of the tangent (excluding the hypotenuse) are the same, and as we are dividing these two numbers, we will get the same answer; 1 for all of them.
Y=x2
Please see graph
As we can see, the gradient is the x value x 2 (g=2x). The tangent method is still accurate so I will use it again for y=x³. I am predicting that the formula for the gradient on y=x³ will be g=3x.
Y=x³
Please see graph y=x3
As you can see my predicted formula did not work. Also, as the lines are getting more and more steeper the tangent method is becoming increasingly inaccurate.
My predicted formula g=3x produces a result which I can use to figure out a formula; every time (excluding the first answer) I multiply x by 3 I get a number, and if I square that number I get the gradient. This also proves that the increment method does work.
So therefore the discovered formula is g=3x2
Therefore, I predict that the formula for y=x4 will be g=4x2
Y=x4
As the tangent method is becoming increasingly inaccurate I am just going to use, the already proven to work, increment method.
Please turn over
As you can see from above, my formula does not work. But there is a pattern somewhere so I am going to persevere with this pattern and look how the formula changed from the start to get to this.
y=x G = y=x
y=x2 G = 2x
y=x3 G = 3x2
y=x4 G = ?
As I can see that in the formula the number which multiplies to the x value increases by 1 each time. Now the x value is always there, and the same so I will leave that. Though the power multiplying the x value is increasing by 1 as well.
So by analysing the pattern the new formula, which hopefully should be successful which I will use for y=x4 will be G=4x3
Y=x4 Retry
Success! I now have a pattern that works, so I can make a formula from this.
The Gradient Function Formula
G = pxp-1
The G is the gradient and p stands for power.
An example of how this formula is used –
I would like to find y=x3
Therefore the formula would be used like this – g = 3x2
To be sure of this formula, I am going to test it using y equals x to the negative 1, y equals x to the minus half and y equals x to the 2x2.
Y=x-1
The method has been successful.
When trying y=x-1/2 I encountered various errors with the calculations so I have decided to leave that one.
More Complex Equations
So far, I have only covered numbers where there is an x to a power. Now to show that the formula works with more complex equations that have more parts than just the xx.
y=2x2
This is more simple than it actually seems. The method for working out this one involves splitting up the equation so that I work out the y=x2 first in the way I have explained with the formula, and then just multiply the output of the formula by the number next to x , which in this case is 2.
y=2x2+3x
This is also the same as above, you split the equation up. I would split up the y=2x2 and work that out using the method above. Then I would take the +3x and work it out as y=3x. After the gradient has been attained for those two parts, I would then add it together and then we would have our gradient.