Table of Contents
1. Introduction
Differentiation is the process where you find the rate of change of a function. The rate of change is sometimes also referred to as the gradient or the tangent, but in differentiation we call the rate of change the “derivative“. This is the gradient of a single point on a graph. The symbol for a derivative is “dy⁄dx“.
Legend
Δ (delta) = Change
∠ = Angle
≈ = Approximately equal to
dy⁄dx = Derivative
2. Linear graphs
With a linear graph (which is just a straight line) it is easier to find the gradient then when we have a non-linear graph. In the example below we have a linear graph in which we use the tangent to find the gradient.
- For a recap on the tangent, visit the post about it here: "Sine, Cosine and Tangent"
- Check if your calculator is set to "degrees" or "radians" depending on the solution needed. You can change this setting by pressing the "mode" button on your calculator until you see "Deg" or "Rad". Select the one you need.
3. dy/dx
To find a gradient, all we need to know are two points on the graph. Divide the “change in y” (opposite side) by the “change in x” (adjacent side) and you get the gradient. That’s also what “dy/dx” stands for. The “d” is short for Δ (delta). Delta means “change”. So “dy” stands for “change in y” and “dx” stands for “change in x”.
4. Non-linear graphs
With a non-linear graph we are going to use the same principle, only in a slightly different way. Because a non-linear graph is curved and not straight, we’ll need to simulate a straight line. We do this by making the adjacent side as small as possible. Basically, we are just zooming in on the graph until it seems like it is a linear graph.
For the non-linear graph below, we have the example function of “f(x)”. The solution of this function gives us the “y” coordinate and the “x” within the function is the “x” coordinate. Because we want the change on the “x” axis to be as small as possible to simulate a linear graph, we make this change as close to zero as possible, something like “0.001”. In the example below this is written as “dx”.
5. Step by step
Step 1: Find the “x” value of the point “A” of which you need the derivative.
Step 2: For the second point “B”, add a change to the “x” value of “A” that is close to “0” e.g. “0.001”.
Step 3: Calculate the “y” coordinates by filling the “x” coordinates in to the function.
Step 4: Calculate “dx” and “dy” with subtraction.
Step 5: Divide “dy” by “dx”.
6. Summary
To find the derivative of a function, you have to divide the change in “y” (dy) by the change in “x” (dx).
“dy” is the same as the change in “y”. This is the opposite side.
“dx” is the same as the change in “x”. This is the adjacent side.
“dy/dx” is the same as “opposite side”/”adjacent side”, which is the gradient (tangent).
To simulate a straight line on a non-linear graph, we make “dx” as close to “0” as possible. Basically, we are just zooming in on the graph until it seems like it is a linear graph.
7. Formulas
Tangent of an angle:
Opposite side / Adjacent side = Tangent of an angle
dy/dx:
(f(x+dx) – f(x)) / dx = Derivative
This Post Has 2 Comments
awesome and simple explanation!
awesome and simple explanation!