Defining Average and Instantaneous Rates of Change at a Point 📍
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An important concept to grasp in this lesson is the difference between the average rate of change and the instantaneous rate of change. The average rate of change is simply the slope of the secant line between two points. The instantaneous rate of change is the slope of the tangent line at any given point (the derivative).
The rate of change formula (pictured below) is the slope of the secant line between two points. "f(b)" represents your y-value to your first point and "f(a)" represents your y-value to your second point, "b" and "a" are the corresponding x-values to those coordinates.
The instantaneous rate of change formula (pictured below) is the slope of the tangent line at a given point. Many people are puzzled with f(x+h) but it simply means that whenever you see an x in your original equation, you insert "x+h." For example, if f(x) = x² + 3x - 9 then f(x+h) = (x+h)² + 3(x+h) - 9. The second part of this formula is subtracting f(x) from f(x+h); however, a common error is that students forget to distribute the "-" sign to all the terms in the original function. The last part is to put everything over h and simplify the entire equation by combining like terms and factoring). If there are any h terms left you evaluate them as 0 and then simplify.
- The formula pictured below is the definition of the derivative.
Defining the Derivative of a Function and Using Derivative Notation 🤓
The function that gives the instantaneous rate of change is the derivative, and the derivative is the slope of the tangent line to the graph at a given point. The formula for the derivative is the same as the instantaneous rate of change formula.
The AP exam loves using the notation for the derivative, so don't be scared as they all mean the same thing; however in certain scenarios in calculus, we may use one notation over the other. Here are some of the ways we can express the derivative (pictured below). For now, only review the notations concerning the first derivative. The bottom line is that y' is synonymous with f'(x) or dy/dx or d/dx f(x)!
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Connecting Differentiability and Continuity: Determining when Derivatives DO or DO NOT EXIST ❗
If you need a complete refresher on continuity you can watch a replay of our stream on Continuity here
Being differentiable means that a derivative exists. It is important to know that being differentiable is being continuous however being continuous does not mean you are differentiable.
On top of being continuous in order to be differentiable, the function must have NO corners, cusps, and no vertical tangent lines.
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Applying the Power Rule 🦸♂️🦸♀️
Using the definition of the derivative for every single problem you encounter is a time-consuming and it is also open to careless errors and mistakes. However, one great mathematician decided to bless us with a fundamental rule known as the Power Rule, pictured below.
For example, in x² your "n" term would be 2. Additionally, you multiply that "n" value to the term's coefficient (in x² your coefficient is 1) and then decrease your terms exponent by 1 (using the power rule the derivative of x² is 2x).
Derivative Rules: Constant, Sum, Difference, and Constant Multiple ➕➖
These first set of derivative rules are simple but absolutely crucial to your understanding of calculus.
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While these two rules may seem confusing, they are actually straightforward. The sum and difference rules are essentially applications of the power rule to every term, as well as combining them (if possible). Here is an example of using both these formulas, pictured below:
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For example, the derivative of 5・x², (5x²) is equal to 5 times the derivative of 2x.
Derivatives of cos, sin, natural exponential functions, and the natural log 🤔
These rules must be committed to memory as they are used throughout the year in calculus.
- The derivative of sin (x) is cos (x).
- The derivative of cos (x) is -sin (x).
If you would like to find a derivative of a trig function with a constant (such as 5sin(x)), you would use the constant multiple rule to get 5cos(x).
- It is important to note that the derivative of these functions only work when you are only using "x" in the function. For example, the derivative of sin(3x) is not cos(3x), in order to get the correct derivative you would need to apply the chain rule. (Do not worry about the chain rule in this unit as it is covered in Unit 3 of calculus.)
The derivative of e^x is e^x. No matter how many times you take the derivative of this function, the derivative of e^x will remain as e^x.
The derivative of ln(x) is pictured below. If you would want to find the derivative of ln(4x), you would need to apply the chain rule.
The Product Rule ✖
By now, we know how to add and subtract derivative functions, but what about multiplying them? With the product rule we can finally multiply derivatives together. Here is the rule (pictured below)
f(x) and g(x) represent two different functions that are being multiplied together. Here is an example of how to apply this rule (pictured below).
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Let's call the first function f(x) and the second function g(x). For our first part, we will take the derivative of the first function and multiply it by the original second function. For the second part, we then multiply the original first function and multiply that by the derivative of the second function. Finally, we will add up both parts.
If it is helpful to remember the derivative of first times second plus derivative of second times first, go for it!
The Quotient Rule ➗
Let's now move on to the product rule's partner: the Quotient Rule! With the quotient rule, we can finally divide derivatives. Here is what the quotient rule looks like. pictured below:
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The quotient rule states that f(x) is the top function (the dividend) and g(x) is the bottom function (divisor). The first part would be to multiply g(x) by the derivative of f(x) and the second part would be to multiply f(x) by the derivative of g(x). After that, you would subtract the two parts (don't forget to distribute the negative sign!). Lastly, you would divide everything by g(x)². If you need another visual, here is an example (pictured below).
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Note that (4x-2) is your top function, f(x) and (x²+1) is your bottom function, g(x).
Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions 🔎
Here is a helpful chart with the derivatives of the rest of the trigonometric functions besides sine and cosine:
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However, it is important to take note that AP Calculus mainly focuses on the derivatives of sin, cos, and tan.
Final Note 🎓
Make sure you get the basics down of unit 2 of AP Calculus AB for this unit sets the foundations of calculus, which is essentially the rest of the course. As a result, it is essential for you to understand these concepts!