I am left with the question of accuracy. I cannot get total accuracy, but there are ways I can get very close to an accurate answer. One such way is to use the method of using a line inside the curve.
(4,16)
(3.5,12.25)
(3.01,9.0601)
(3,9)
Judging by this trend, I can use this method to find more accurate results. If I were to use very short lines, (3,9) to (3.0000001,9.0000006) for example, then use the y step/x step method to find the gradient, I would get an answer very close to the correct gradient. In this case, I would get a gradient of 6 for the tangent at x=3, which from the graph work I know to be a correct answer.
I will use this method to find the gradients for the graph of y=x3. I will make the difference between x1 and x2 0.0000001. The values in the following table are rounded.
The first numbers I saw were the first and third numbers. I noticed these because the first number was 3, the power by which I was multiplying the values. The third number was 27, which is 33. This struck me as strange - only this number was cubed, none of the others. To explain this, I tried cross analysing these results with the results I obtained from the y=x2 graph. I saw that the first value was the value of the power by which I multiplied the values. The third value however was not 32, but the second value was 22. Moving back to the results for y=x3, I tried dividing all the results by 3, and was left with the x1 values squared. Therefore, the formula for the gradient of a tangent on the graph of y=x3 was 3x2. Looking at the similarities between the formula for x2 and x3, I saw that the formula forx3 was the same as the other formula, but displaced by one. To comply with this statement, a way of writing the first formula would be 2x1. You can see by the modified version of this formula that if you added 1 to every numerical value in the formula, you would have the formula for the graph of x3. I can predict from this trend that the formula for the gradients of the tangents on the graph of x4 will be 4x3. To check this I will work out the gradients for the tangents of the points 1-6, having again a difference of 0.0000001 between the values of x1 and x2. I predict the values for (y2-y1)/(x2-x1) will be if not the same then very near the results given by the formula. The values in the following table are rounded.
As I thought, the two sets of results are very close. This shows that the formula is correct. I now need to find a formula which link the different formulae together. As the first number in the formulae is always the power being multiplied by x, and the power in the formulae is always 1 less I can determine the linking formula.