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published on august 5, 2020
Last updated on August 4, 2020
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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 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 me Tan 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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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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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.
Your "n" term is the power your x-term is being raised to.
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 product rule the derivative of x² is 2x).
Remember: the derivative is the slope of the line tangent to any point on the graph and using the power rule (like the definition of the derivative) simply gives you the formula to the slope of the line tangent. If you would like to know what that point is then you would have to evaluate using your x-coordinate.
These first set of derivative rules are simple but absolutely crucial to your understanding of calculus.
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The sum rule states that the derivative of a sum of functions is the same as the sum of their derivatives.
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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The constant multiple rule states that the derivative of a constant times a function is equal to the constant times the derivative.
For example, the derivative of 5・x², (5x²) is equal to 5 times the derivative of 2x.
The constant rule states that the derivative of any constant is 0 (for example, the derivative of 5 is 0).
These rules must be committed to memory as they are used throughout the year in calculus.
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).
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 e^x only works when it is raised to only the "x" power. For example, the derivative e^2x is not e^2x, in order to get the correct derivative you would need to apply the chain rule.
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.
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!
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).
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.
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!
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