The Linear Properties of a Derivative

The Linear Properties of a Derivative
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  • 0:06 Quick Review of Derivatives
  • 1:06 Constant Rule of Derivatives
  • 4:48 Distributive Type Rule…
  • 7:56 Lesson Summary
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Lesson Transcript
Instructor: Zach Pino
In this lesson, learn two key properties of derivatives: constant multiples and additions. You will 'divide and conquer' in your approach to calculating the limits used to find derivatives.

Quick Review of Derivatives

The derivative of a function is its rate of change.
Review of Derivatives

The derivative of a function is the rate of change of that function. So for something like y=f(x), the derivative is how y is changing with respect to x. Formally, we write this as y`=dy/dx, and that equals the limit as delta x goes to zero of (f(x + delta x) - f(x)) / delta x. Now some derivatives are just plain hard. So can we use what we know about the properties of limits - what we know about this part - to make finding the derivative easier? Well for limits, we know to 'divide and conquer'. The two big properties of divide and conquer that we can use here are properties for constants and distributive type properties.

Constant Rule of Derivatives

Let's take a look at these in turn. The first property that we're going to look at is regarding constants. Let's do this with an example. Let's say that f(x)=60x^2. The derivative of f(x) is f`(x), and I'm going to write this as d/dx of 60x^2. This is how 60x^2 is changing with respect to x. Formally, this is the limit as delta x goes to zero of (f(x + delta x) - f(x)) / delta x. Let's plug in f(x) and f(x) + delta x. So we have the limit as delta x goes to zero of (60(x + delta x)^2 - 60x^2) all divided by delta x. We can expand this and simplify, knocking out the 60x^2s. We can divide both the top and the bottom of this by delta x. What we find is that f`(x) is the limit as delta x goes to zero of 120x + 60delta x. And that just equals 120x. So let's keep that in mind. We just found, formally, that f`(x)=120x.

In the first example, simplify by canceling the 60x^2s and dividing the top and bottom by delta x.
Constant Rule Example 1

Now let's back up a few steps. So we had the limit as delta x goes to zero of (60(x + delta x)^2 - 60x^2) all divided by delta x. Now I could pull out the 60. If I pull out the 60, I have the limit as delta x goes to zero of 60 times this bigger fraction. Now I've got the limit of one function - 60 - times another function - this ((x + delta x^2) - x^2) / delta x. I know that with properties of limits, I can divide this and conquer. I can write this as the limit as delta x goes to zero of 60 times the limit as delta x goes to zero of this other mess. Well, the limit as delta x goes to 60 is 60; it's the limit of a constant. So I can calculate this out. Now the limit as delta x goes to zero of ((x + delta x^2) - x^2) all divided by delta x can be found by first multiplying out this term, crossing out the terms and simplifying and dividing the top and bottom by delta x. I find that this limit is equal to 2x, So the whole thing is, again, 120x, which is exactly what we found before.

What does this tell us? This says that if you have a function like y=Cf(x), where C is a constant, like 4, you can pull the constant out. When you're finding y', that's taking the derivative (d/dx) of this entire thing (Cf(x)), and then you can pull out the C. So y`= C * d/dx * f(x). So if you have a function, y=Cf(x), the derivative of that function is y`=Cf`(x). This is our first rule, for constants.

Distributive Type Rule of Derivatives

The derivative of y = 2x is just 2.
Distributive Rule Example 1

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