Calculus sounds really difficult. Really it is fairly simple mathematical tool. It stems from a really simple algebraic form, the slope of a curve. All of those who have taken algebra are familiar with the slope intercept formalization of a linear curve where a is the slope coefficient and b is the y-intercept.
The mild complexity arrives with curves that are not straight. Something like:
Without getting bogged down in infinitesimals, it is fairly clear if you think about it that the slope of this curve is different at every point x. When a value is different for ever given value x, it too is a curve. This curve is the derivative. In this case it is:
You can see that where the slope is horizontal in the original curve at x = 0, the derivative is at y = 0. The derivative is just a graph of the slope of a curve. It is just a continuation of the slope intercept formalization for non-linear curves.
Some readers may have noticed that the y-intercept is not retained by the derivative. This is true. This is why when taking the integral (i.e., the opposite operation from the derivative), mathematicians always add back a constant C. This constant is the y-intercept of the original curve. Derivatives represent only the slope and so lose this value. Using the constant makes it clear that this value was forgotten in the process of derivation.
Calculus is concerned quite completely with the slope, or a. And indeed the derivative of this formula would be:
The mild complexity arrives with curves that are not straight. Something like:
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| Source: desmos.com |
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| Source: desmos.com |
You can see that where the slope is horizontal in the original curve at x = 0, the derivative is at y = 0. The derivative is just a graph of the slope of a curve. It is just a continuation of the slope intercept formalization for non-linear curves.
Some readers may have noticed that the y-intercept is not retained by the derivative. This is true. This is why when taking the integral (i.e., the opposite operation from the derivative), mathematicians always add back a constant C. This constant is the y-intercept of the original curve. Derivatives represent only the slope and so lose this value. Using the constant makes it clear that this value was forgotten in the process of derivation.
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