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...
Of course, derivatives are not going to always be as easy as finding the gradient of a straight line.
To find a derivative, you could use the formula seen in the last chapter, ∆y÷∆x, or you could just look at a graph showing the line, and see how big the slope is, but that only works for straight lines (linear equations).
To work out a derivative of a curved line, let's have a look at the most basic curved formula: x squared.
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That looks pretty fun, doesn't it? Actually, that might just be me, but the real fun happens when we zoom in on the curved part. Specifically, on the part marked (1,1).
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x squared is a curved line, but when you zoom in on it, it becomes straight. Another way of saying this is, the smaller the increase in x, the straighter the line. Remember, the function of a straight line is mx+c, and the derivative is just m, or the gradient. Let's pretend for a second that the graph starts at x=0.9998. 0.9998 squared is equal to roughly 0.9996. 1 squared is equal to exactly 1. To find the derivative, we need to find Δy and Δx. The increase in x is 0.0002, and the increase in y is 1-0.9996, or 0.0004. Δy/Δx=0.0004/0.0002=2
Now let's look at a different part of the graph, (2,4).
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1.9999 squared is equal to roughly 3.9996, and 2 squared is for. Like before, we simply find Δy and Δx, and divide. The increase in y is equal to 4-3.9996, or 0.0004, and the increase in x is equal to 2-1.9999, or 0.0001. 0.0004/0.0001 is equal to 4, so the derivative at this point on the graph is equal to 4.
Now, how does the derivative of the graph change? Is that even possible? Well, no. Not technically. But for curved lines, we need to find a stable derivative "with respect to x". This means that the derivative changes throughout the graph, but as a whole, it is only one derivative. I think I mentioned it before, but dy/dx literally means "the derivative of y with respect to x". In x squared, the derivative at x=1 is 2. The derivative at x=2 is 4. For your information, the derivative at x=3 is 6, and so on. The derivative is always two times larger than the x-value. We could say that the derivative of y with respect to x is 2x. This works with any function.
