Where the derivative comes from, how to estimate a limit without any algebra, and what a rate of change actually tells you.
You can find the gradient of a straight line because you can pick two points on it. On a curve, every pair of points gives a different answer, and none of them is the gradient at a point.
So watch what happens when the two points move together. P is fixed at (1, 1). Q slides down the curve towards it. Press play.
Drag the slider yourself, or press play. Nothing is stored and nothing is sent anywhere.
This is the whole method. Shrink h, write down the gradient, and see what the numbers are heading towards. Rows fill in as the animation passes them.
| h | Q | gradient of PQ |
|---|---|---|
| 1 | (2, 4) | – |
| 0.5 | (1.5, 2.25) | – |
| 0.1 | (1.1, 1.21) | – |
| 0.01 | (1.01, 1.0201) | – |
| towards 0 | towards P | – |
The gradients are 3, 2.5, 2.1, 2.01. They are not reaching 2 by accident: for this curve the gradient of PQ is exactly 2 + h, so as h shrinks the answer is squeezed towards 2.
That number, 2, is the gradient of the tangent at P, and it is what we mean by the derivative at x = 1.
You are not asked to find limits by algebra on this course. You are asked to estimate one from a table or a graph, which is exactly what you just did. If a question gives you a table of values and asks what the gradient is approaching, that is the whole task.
Examiners switch between these freely, and a question about a balloon will not use the letter x. They all mean "the rate at which the top variable changes when the bottom one changes".
| Written | Say it as | Used when |
|---|---|---|
| dydx | dee y by dee x | y is a function of x |
| f′(x) | f prime of x | you were given f(x) |
| dVdr | dee V by dee r | volume changing with radius |
| dsdt | dee s by dee t | distance changing with time |
This is the half of 5.1 that carries the marks, and it has nothing to do with limits.
A workshop's cost in baht for making x chairs is C(x) = 0.02x² + 5x + 200, and its derivative is C′(x) = 0.04x + 5.
So at 100 chairs, making one more chair adds about 9 baht. That sentence is the answer. "The derivative is 9" is not, because it does not say nine of what, per what.
Units come free, and they are marks. The units of a derivative are always the top units divided by the bottom units. Baht per chair. Litres per second. Marks per hour of revision. If you can say the units you usually have the interpretation.
A spherical balloon has volume V = 43πr³, and dVdr = 4πr².
When the radius is 3 cm, the volume is growing at about 113 cm³ for every extra cm of radius. Notice it is not constant: the same balloon at r = 6 grows four times as fast, because the derivative has an r² in it.
1. A table shows the gradient of a chord as 4.3, 4.1, 4.02, 4.001 for shrinking h. Estimate the gradient of the tangent.
2. For the chair workshop, C′(100) = 9. What does that mean?
3. For the balloon, dVdr = 4πr². Find the rate when r = 3, to one decimal place.
Reading a limit off a table is one mark and almost free, once you know that is what is being asked. Do not try to do algebra to it.
Interpreting a derivative in context with units is the mark people lose. "9" earns nothing. "At 100 chairs, one more chair costs about 9 baht" earns it.
Watch the letters. A question about a cone will ask for dV/dh or dV/dr, and they are different derivatives of the same formula. Read which variable is changing before you differentiate anything.
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