Lesson 2 — Differentiation | Calculus 1/AP Calculus AB Condensed Course
Definition of a Derivative
A derivative is the rate of change or slope of a function at a single point in time. It’s useful in modelling the change of a complex function with a non constant slope. The mathematical definition of it is as follows:
Both of these limit expressions are both the same but in different notation. The upper expression uses the slope equation Δy/Δx and takes the limit as h goes to 0, where h is the x-distance between f(x+h) and f(x); this makes sense because as the x-distance between the two points gets closer and closer, we get the slope at a single point. The lower expression also uses the slope equation but instead, it takes the limit of x approaching a, which gets it infinitely close to x=a and f(a), creating a distance of 0 and calculating the slope at x=a.
One thing to note about derivatives is that if a function f(x) is differentiable, meaning its derivative exists at all input values, then it implies it is continuous. This is because a slope existing at all points means that there are no sudden jumps or breaks in f(x).
Concavity
Concavity is the state of a function where its derivative is either decreasing or increasing. In other words, it indicates the sign of the second derivative. There are 2 types of concavity: positive/up and negative/down. Concave up means the derivative is increasing and the second derivative is positive. Concave down means the derivative is decreasing and the second derivative is negative. The second derivative is the “derivative of the derivative”, or as I like to think of it, the “slope of the slope”.
When I think concavity, I always think quadratic functions. Picture the graph of x²; as x increases, the slope seems to increase really fast. If we picture -x², the slope seems to decrease really fast. After knowing the methods of calculating derivatives in the next section, you’ll see that the second derivative of x² is always positive and that of -x² is always negative.
Calculating Derivatives
credit: https://coachforrester.weebly.com/uploads/1/3/1/9/13191763/ap_calculus_ab-bc_cheat_sheet.pdf
A fundamental derivative formula missing here is power rule: the derivative of f(x) = x^n is f’(x) = nx^(n-1).
The prime notation f’(x) indicates the first order derivative of f(x), f’’(x) indicates the second order derivative, and so on. You just need to memorize these rules and practice using them until they become part of you.
Note: For chain rule, the function g(x) is just a “subfunction” of f(x). For example, (1-x)³ has the subfunction g(x) = 1-x and the main function f(a) = a³ where a = g(x). The names of the functions g and f are arbitrary.
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Derivative for an inverse function where f(x)=g⁻¹(x):
credit: https://graphicmaths.com/pure/differentiation/inverse-function/
The proof for this is quite simple, so it may be worth understanding. If we look back at f(x)=g⁻¹(x), we can note that g(f(x)) = x using inverse multiplication. Here, we can differentiate the function using power rule on the right side and chain rule on the left side of the equation. After this, we would get g’(f(x))f’(x) = 1, and we can solve for f’(x) here.
Important Derivative Identities
credit: https://magoosh.com/hs/ap/common-derivatives-ap-calc-exam/
credit: https://www.geeksforgeeks.org/maths/differentiation-of-inverse-trigonometric-functions/
The proof for each identity is too long so it is more efficient to just memorize them.
Implicit Differentiation
Implicit differentiation is calculating the derivative of both sides of an equation. This is typically done using an equation involving an output of y and input of x. For example, if we differentiate y²-yx = 2x with respect to x, we will get 2ydy/dx-xdy/dx-y = 2. This method of differentiation is helpful when we cannot isolate y on one side of the equation. dy/dx is the other notation for the derivative, resembling the slope equation of Δy/Δx.
Many get confused with why, for example, the derivative of y² is 2ydy/dx. We are using the power rule in this case, however we are differentiating with respect to (or with a changing input) of x. We are accounting for the fact that the output y is dependent on the input x. Anyways, this seems only like power rule but it’s really just chain rule in action. We can see that the base function g(a) = a² and the main function a = f(x) = y, so y² = g(f(x)). Keep in mind that f’(x) = dy/dx. When we differentiate this with respect to x, we get g’(f(x))f’(x) = 2a(f’(x)) = 2f(x)f’(x) = 2ydy/dx.
L’Hopital’s Rule
credit: https://calcworkshop.com/derivatives/lhopitals-rule/
If the second iteration of this rule doesn’t work, keep applying the rule until you get a definite answer.
