Calculus surrounds all of us every day. But most people either don't want to learn about this glorious subject, or find it too difficult. That is where this book comes in. It will explain what calculus is, and include detailed examples to help you u...
In the world of normal people (sort of), a derivative is more commonly known as a gradient. You may have learned in primary school that the gradient can be defined as "the increase of the y-value as the x-value increases by 1", but, just as everything else changes when it gets to Calculus, the gradient changes as well. And that is speaking literally. In the line of x^2, the gradient changes from 2 to 4 to 6 to 8 and so on. Before we delve too deep into differentiation, we need to learn about a really cool Greek letter known as ∆, or δ. Unfortunately, to save time, I am using a key board to write this, and it doesn't have the ability to write the lower case squiggly form thingy, so I will be using the uppercase when writing, but lowercase in equations. By the way, it is called delta. Another way to find the gradient of a straight line is to use the following equation:
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As you can see on the graph, there is a dot on point (1,7), which is a substitute of (x₁,y₁), and a point on (3,11), which is a substitute for (x₂,y₂). Using the above equation, we can work out that the gradient is:
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There is another way to write this though:
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The delta symbol means "a small increase". So the expression is saying, "a small increase in y divided by a small increase in x". Generally, ∆ works best when very small. If my existing explanations of ∆ have not been helpful, imagine this: a line has an x-value of 4. Then, later in the line, it has an x-value of 19. This means that ∆x is equal to 15, as 19-4=15. On a graph, every x-value has a corresponding y-value. Likewise, every ∆x-value has a corresponding ∆y-value.
But what does this all have to do with derivatives? Easy. Derivative is just a fancy word for gradient.
So the derivative of the straight line above is 4.
