Differentiation is the process of finding the gradient function. To differentiate a curve, you can draw a tangent to the curve at a certain point. Then by using the equation dy / dx, we can find the gradient of the curve. The smaller the values, the more accurate the results, but it is inefficient and inaccurate.
To differentiate equations: y = ax2 + bx + c:
e.g. y = 2x2 + x + 2, dy / dx = 4x + 1.
When a function of x, y = f(x), is differentiated once, the result is called:
Similarly, when a function of x, y= f(x), is differentated twice or three times, it is called the same, but the word 'first' is replaced by the corresponding amount. The gradient of the gradient function is when the function has been differentiated twice, i.e. d2y / dx2. The third is d3y / dx 3
The normal is the line which is perpendicular to the tangent. If two lines are perpendicular, their gradients multiply to equal -1. To find the equation of a line which passes through a certain point, for example a tangent, this formula can be used:
You are trying to find y and x. x1 will be given to you and so will y1 (because you can put x into the value of y given to you. E.g. y=2x2 - 3x, where x = 1. You can work out that y = -1). Then to get m, you need to differentiate the original y value. Then you can use the formula to find the equation of the tangent. Similarly, to find the equation of the normal, - 1 / m = -1. So you can use this new value of m1 to put in place of m in the formula. For example:
Find the equations of the tangent and the normal to these curves at the points corresponding to the given values of x:
y = 3x2 + 2, x = 4.
y1 = (16 X 3) + 2 = 50.
dy / dx of y = 6x. The gradient, m, therefore = 6 X 4 = 24
Using the formula: y - 50 = 24(x - 4). Therefore, y = 24x - 46.
Now we can find the normal: If m = 24, then m1 = - 1/24
y - 50 = - 1/24x + 1/6, therefore y + 1/24x = 50 1/6. Or 24y + x - 1204.
This means the integral of xn with respect to x.