#Principal definition math how to

We write this as dy/dx and say this as вЂњdee y by dee xвЂќ. We can do this calculation in the same way for lots of curves. Note that when x has the value 3, 2x has the value 6, and so this general result agrees with the earlier result when we calculated the gradient at the point P(3, 9). вЂlimвЂ™ stands for вЂlimit вЂ™and we say that the limit, as x tends to zero, of 2x+dx is 2x. We have a concise way of expressing the fact that we are letting dx approach zero. So the coordinates of Q are (x + dx, y + dy).Д«ecause we are considering the graph of y = x 2, we know that y + dy = (x + dx) 2.ДЄs we let dx become zero we are left with just 2x, and this is the formula for the gradient of the tangent at P. The corresponding change in y is written as dy. The x coordinate of Q is x + dx where dx is the symbol we use for a small change, or small increment in x. Point Q is chosen to be close to P on the curve. We will now repeat the calculation for a general point P which has coordinates (x, y). Observe that as Q gets closer to P the gradient of PQ seems to be getting nearer and nearer to 6. The gradient of the line PQ, QR/PR seems to approach 6 as Q approaches P. The table below shows the effect of reducing PR successively, and recalculating the gradient.

The gradient of PQ will be a better approximation if we take Q closer to P. We can take the gradient of PQ as an approximation to the gradient of the tangent at P, and hence the rate of change of y with respect to x at the point P. Knowing these values we can calculate the change in y divided by the change in x and hence the gradient of the line PQ. The x coordinate of Q is then 3.1 and its y coordinate is 3.1 2. Suppose we choose point Q so that PR = 0.1. Point R is vertically below Q, at the same height as point P, so that в–іPQR is right-angled. We will choose Q so that it is quite close to P. We choose a nearby point Q and join P and Q with a straight line. The graph below shows the graph of y = x 2 with the point P marked. So if we calculate the gradient of one of these lines, and let the point Q approach the point P along the curve, then the gradient of the line should approach the gradient of the tangent at P, and hence the gradient of the curve.Д®xample : We shall perform the calculation for the curve y = x 2 at the point, P, where x = 3. The lines through P and Q approach the tangent at P when Q is very close to P. We see that the lines from P to each of the QвЂ™s get nearer and nearer to becoming a tangent at P as the QвЂ™s get nearer to P. We also show a sequence of points Q1, Q2. We use this definition to calculate the gradient at any particular point.Д¬onsider the graph below which shows a fixed point P on a curve. NOTE: The gradient of a curve y = f(x) at a given point is defined to be the gradient of the tangent at that point. The rate of change at a point P is defined to be the gradient of the tangent at P. This is defined to be the gradient of the tangent drawn at that point as shown below

#Principal definition math how to

We now explain how to calculate the rate of change at any point on a curve y = f(x). The rate of change of y with respect to x is not a constant.Д¬alculating the rate of change at a point

So even for a simple function like y = x 2 we see that y is not changing constantly with x. But when x increases from в€’2 to в€’1, y decreases from 4 to 1. Note that as x increases by one unit, from в€’3 to в€’2, the value of y decreases from 9 to 4. Joining different pairs of points on a curve produces lines with different gradients NOTE: For a straight line: the rate of change of y with respect to x is the same as the gradient of the line.Дifferentiation from first principles of some simple curvesДЇor any curve it is clear that if we choose two points and join them, this produces a straight line.ДЇor different pairs of points we will get different lines, with very different gradients. Observe that the gradient of the straight line is the same as the rate of change of y with respect to x. We say that вЂњthe rate of change of y with respect to x is 3вЂќ. In other words, y increases as a rate of 3 units, for every unit increase in x. Look at the table of values and note that for every unit increase in x we always get an increase of 3 units in y. Values of the function y = 3x + 2 are shown below No matter which pair of points we choose the value of the gradient is always 3. When x changes from в€’1 to 0, y changes from в€’1 to 2, and so We take two points and calculate the change in y divided by the change in x. We can calculate the gradient of this line as follows. This section looks at calculus and differentiation from first principles.ДЄ straight line has a constant gradient, or in other words, the rate of change of y with respect to x is a constant.Д¬onsider the straight line y = 3x + 2 shown below