Calculus... one of the greatest inventions of mathematics. It is the study of change and is a fundamental concept that has been 'integrated' into the natural world. :P
From projectile motion, to economy, to population gro...
But this is the gradient of a line created by two points. How do you find the gradient of a tangent at one point? Well, if you decrease the distance of x + h and you make it get infinitely closer to x, you end up with one point right? That one point being at x. How do you do this? You decrease the value of h, to make the distance smaller. In fact, you decrease the value of h until it becomes zero. Thus we take the limit of h as it approaches 0, and when we do so, we finally get the derivative. (Remember that the derivative is the equation that tells you the gradient of a tangent at a point. For now, it is denoted as f'(x))
The equation here sums it up.
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Remember our parabola from before? We can now find its derivative properly, using First Principles. From what we know, the function of the parabola was f(x) = x²+1.
Therefore, we use the limiting equation we have above, replacing f(x) with x²+1 since f(x) = x² + 1 is the function we are differentiating.
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Expanding the numerator gives,
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Then simplifying,
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We can then factorize h from the numerator.
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The h in the numerator and denominator divide and cancel out. This gives,
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Since h approaches zero, we can now substitute h as 0 into the equation. Once we do so, we prove the derivative of the function f(x) = x²+1.
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The derivative is 2x, and as shown in the previous chapter, this can be used to find the gradient of a tangent at any point.
Now we can see that by using First Principles and the limiting process, we can find the derivative of a function. In fact, you can use this on f(x) = 5x + 8, f(x) = 3x²-2x+18 or f(x) = 6x³, or virtually any polynomial function. The expanding and algebra can get very complicated though, especially with polynomials with higher powers.
That is why, from First Principles, we derive simpler formulas for differentiating all sorts of functions.
The First Principles are the fundamentals of differential calculus. From First Principles, we can derive simpler formulas for differentiating polynomial functions such as f(x) = x² + 1, and later on, it will show us how to differentiate trigonometric, logarithmic, and exponential functions.
In the next two chapters, we will take a look at the other notations for the derivative and introduce the general formula for finding the derivative of polynomial functions in a more efficient manner.
