Introduction
Derivatives are functions that are (as you can assume) derived from other functions. Derivatives tell you the slope (a.k.a. rate of change) of a function at a given point.For example, assume we have a function f(x)=x33:
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The derivative of this function is f′(x)=x2. Notice the ′ after the f. This is one of the ways you can denote a derivative. It's read 'f prime of x'.
We can graph the derivative on top of f(x) to get an idea of what is going on:
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If we compute f′(1) we see get the result of 1. This means that the slope at f(1)=1. You can easily tell if the slope is 1 if it appears to make a 45∘ turn from either positive axis. Likewise, f′(1)=−1, therefore, the slope at f(−1) is -1.
You can compute f′(x) for any x in the domain of f.
Definition
The simplest definition of a derivative involves computing a limit.
Imagine that we want to get the tangent line of the curve f(x)=x33 at x=1. We could start by making an approximation by drawing a line through two points on the curve. those points are at a distance h on the x-axis.
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If we would infinitely keep making h smaller, we would eventually get the tangent line at the point x=1. If we can get the slope of the tangent line (which measures the instantaneous rate of change at a point), we have the derivative. This is the limit definition of the derivative:
limh→0=f(x+h)−f(x)h
Example:
Find the derivative of f(x)=4x2.
Start with the limit definition of the derivative.
f′(x)=limh→0f(x+h)−f(x)h
Substitute.
f′(x)=limh→04(x+h)2−4x2h
Expand numerator.
f′(x)=limh→04x2+8xh+4h2−4x2h
Subtract.
f′(x)=limh→08xh+4h2h
Divide.
f′(x)=limh→0(8x+4h)
Solve the limit.
f′(x)=8x
The limit definition of the derivative is one of the building blocks for some shortcuts for finding derivatives more quickly. I will write more posts concerning derivatives soon. I hope this was useful. All feedback is welcome!
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