These are my complete notes for Differential Calculus, covering such topics as Limits, Squeeze Theorem, Velocity, Acceleration, and Position, the Power Rule, Chain Rule, Quotient Rule, Particle Motion, Continuity, Optimization, Related Rates, and more.
I color-coded my notes according to their meaning - for a complete reference for each type of note, see here (also available in the sidebar). All of the knowledge present in these notes has been filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes and online courses. In the unlikely event there are any egregious errors, contact me at jdlacabe@gmail.com.
Summary of Differential Calculus (Complete)
?. Limits.
# Average Rates/Speed: Average Speed is found by dividing the distance covered by the elapsed time.
# Limit: The value a function approaches as it becomes closer to a specified point - thus, not necessarily actually being at that point, but rather as close to it as possible with being on it. If a function is continuous near that point, the limit equals the function's value; otherwise, it will describe the behavior of the function even if undefined at that point.
# Properties of Limits:
Similar to Logarithms, Limits have their internal rules that can applied at any time. For the "commandments" version of this list, see Rule 110.
1. The limit of a constant is equal to a constant. $$\lim_{x \to a} c = c$$
2. The limit of x as x approaches a, equals a. $$\lim_{x \to a} x = a$$
3. The limit of a sum is the sum of the limits. $$\lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$$
4. The limit of a difference is the difference of the limits. $$\lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$$
5. The limit of a constant times a function is the constant times to limit of the function. $$\lim_{x \to a} [c × f(x)] = c \lim_{x \to a} f(x)$$
6. The limit of a product is the product of the limits. $$\lim_{x \to a} (f(x) × g(x)) = \lim_{x \to a} f(x) × \lim_{x \to a} g(x)$$
7. The limit of a quotient is the quotient of the limits, provided the denominator does not equal zero. $$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$$
8. The limit of a sum is the sum of the limits. $$\lim_{x \to a} (f(x))^n = (\lim_{x \to a} f(x))^n$$
# Rule 106. The average rate of change IS the slope. The average speed is the distance covered divided by the elapsed time, like miles per hour. Rate of increase is also just the derivative. First rate is first derivative, rate of the rate is second derivative, etc.
# Rule 107.Say that 16t² is the free fall motion of an object with no air resistance. It will follow the same (∆y / ∆t) formula as the slope formula.
∆ (Delta) means as "the change in", so (change in distance / change in time). From 0 to 2 seconds, the rock will travel (6(2)² - (6(0)^2)) / (2 - 0), or 34 ft/sec average speed.
# Rule 108. To find the instaneous speed/rate of change, the limit formula is as follows: $$\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$ You know a and the equation at first, plug in (a + h) to the equation and complete formula. Then, plug in 0 for h to find instantaneous velocity/speed.
# Rule 109. Speed is the time rate at which an object is moving along a path, while velocity is the rate and direction of an object's movement.
# Rule 110. There are nine holy truths of limits that must always be remembered and and held immutable. They are similar to the mystical factoring rules, and some are versions of the Limit Properties referenced previously:
- The equivalance of constants. (L.P. #1)
- The equivalance of a to x approaching a. (L.P. #2)
- The Sum Rule. (L.P. #3)
- The Difference Rule. (L.P. #4)
- The Constant Multiplication Rule. (L.P. #5)
- The Limit Product Rule. (L.P. #6)
- The Limit Quotient Rule. (L.P. #7)
- The Limit Power Rule. (L.P. #8)
- The Radical Rule. (a synthesis of L.P. #8 & #7 or #6)
# Rule 111. The one-sided limit is allowed to be infinity, while two-sided limits cannot be infinity and must not be a number.
# Rule 112. The Squeeze Theorem:
When the limit can't be found directly, find it indirectly using the squeeze theorem. The way this theorem works, is to 'squeeze' function f (which has the uncertain limit) between two functions, h and g, and find their limits. Definitionally, this is written as follows:
If g(x) ≤ f(x) ≤ h(x), where x ≠ c in some interval around c, then: $$\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$$
The classic example of a function that must use the Squeeze Theorem is one that oscillates, such as x² × sin(1/x).
The key idea of the Squeeze Theorem is that if h(x), which is above f(x), has its limit at c with the same y-value as g(x), then f(x) would also have to have that limit because it is between g(x) and h(x).
# Infinity/∞: The term 'infinity' does not refer - it describes the behavior of a function. Infinity means an increasing distance forever.
# Rule 113. The squeeze theorem also works for infinite limits.
# Rule 114. To find the horizontal asymptote, you can take the biggest thing on top, and divide it by the biggest thing on the bottom, or you could factor out the biggest base (with exponent), even if it makes every divisor weird. Then, factor to find the vertical asymptote, just plug in zero for the bottom, or cheat by using a graph.
# Rule 115. End Behavior Models
You can determine the end behavior of a complicated polynomial by looking at a similar one that acts exactly the same for extremely large values of x. For all polynomials, g(x) = an × xⁿ is the end behavior.
The function g must be:
a) A right end behavior model for f if $$\lim_{x \to ∞} \frac{f(x)}{g(x)} = 1$$
b) a left end behavior model for f if $$\lim_{x \to -∞} \frac{f(x)}{g(x)} = 1$$
# Rule 116. If the EBM is a constant, then it is a horizontal asymptote.
# Rule 117. A function's right & left end behavior models are not always the same function.
# Rule 118. Tradition dictates that for an equation with only two polynomials, the left one is the REBM and the right is the LEBM. Yes, opposite of what one may immediately think.
# Reciprocal Substitution:
There is a very specific and unique characteristic of Limits that must always be kept in mind when performing problems. Reciprocal substitution can be performed by taking the variable of the limit (90% of the time being x), flip it to its reciprocal (e.g., 1/x for x), and switch the term of the limit as shown below. This reciprocal is performed even for x's inside of trig. functions - it is done for all instances.
$$\lim_{x \to ∞} \frac{1}{x} = \lim_{x \to 0^+} x$$ $$\lim_{x \to -∞} \frac{1}{x} = \lim_{x \to 0^-} x$$
Infinity goes to Zero-Right, as Negative Infinity goes to Zero-Left. The nature of the reciprocal is not specific a specific side - either infinity or the zero could have 1/x or x.
# Rule 119. Rampant idiocracy has softened testing policy enough that you may be able to, in essence, cheat and look at the graph to determine values like the end behavior mode and asymptotes.
# Rule 120. If the assumed number for the LEDM of some two-polynomialed equation doesn't work, then try the answer for the REBM. Same for the issue vice-versa.
# Rule 121. For piecewise functions, even if x is ≥ or ≤ or > or whatever to 0, a (Lim x→0⁻) will make the function go to the f(x) for less than zero, and (Lim x→0⁺) makes it go to x > 0. Think of it like the plus and negative signs -0.1 or add 0.1 to the value.
?. Continuity.
# Rule 122. Any function y = f(x) whose graph y = f(x) can be sketched in one continuous motion without lifting the pencil is an example of a continuous function.
# Rule 123. There are two RULES of CONTINUITY, which if both met, establish UNIVERSAL CONTINUITY for a function.
- A function y = f(x) is continuous at an inerior point c of its domain if:
$$\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$$ $$\lim_{x \to c} f(x) = f(c)$$ - A function y = f(x) is continuous at a left endpoint a if:
$$\lim_{x \to a^+} f(x) = f(a)$$ and is continous at a right endpoint b of its domain if: $$\lim_{x \to b^-} f(x) = f(b)$$ If a function f is not continuous at a point x, we say that f is discontinuous at c, and that c is a "point of discontinuity" of f.
# Rule 124. Continuous Function are continuous everywhere on their domains. Properties of Continuous Functions f & g are continuous at x = c.
All polynomial and Radical functions are continuous everywhere with no Points of Discontinuities. On the other hand, all rational and trig. functions are only continuous along their domain; they have Points of Discontinuities wherever the denominator equals 0.
# Operation Terminology Refresher:
- Sums: f + g
- Differences: f - g
- Products: f × g
- Constant Multiples: k × f, where k = any real #.
- Quotients: f/g, g ≠ 0.
# Rule 125. If f is continuous at c, and g is continuous at f(c), then the composite g ∘ f is continuous at c.
# Rule 126. A function is continuous even if it has Points of Discontinuity, as long as these points are not within the domain. For example, 1/x is considered to be continuous along its domain, despite the absurdity of its graph.
# Rule 127. A function passes the Intermediate value theorem if it never takes on two values without taking on every value in between. A continuous function f on a closed interval [a, b] will take on every y-value between f(a) and f(b), given no discontinuities.
# Rule 128. Jump Discontinuities: Where the left & right side limits are not equal.
Infinite Discontinuities: Where the discontinuity is an asymptote.
Removable Discontinuities: A detached dot, floating on the plane, where the limit continues to be a real number.
Oscillating Discontinuities: A discontinuity that oscillates back and forth, son. Examples include sin(1/x). These functions are discontinuities because no matter how far you zoom in, it will still be oscilliating, thus lacking local linearity.
# Rule 129. For Discontinuities: Jump beats hole. Infinite beats Jump. Oscillating is in a league of its own, as it can only be sin(1/x) or something of the sort.
?. Derivative.
# Rule 131. A tangent line is a line that intersects a curve at exactly one point. The slope of the tangent line is the instantaneous rate of change at a point.
# Difference Quotient: $$\frac{f(x + h) - f(x)}{h}$$ The "Difference Quotient", given above, represents the secant line of a curve from the point (a, f(a)) to (a+h, f(a+h)), which is also the average rate of change from x=a to x=a+h:
# Normal Line: The line normal to a curve is the line perpendicular to the tangent at that point.
# Rule 132. When the limit exists, it's called the derivative of f at a, or any point. The derivative is f'(x) = $$\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$ provided the limit exists. "Differentiate" means to determine the Derivative.
# Verbal Descriptions of Calculus Notation:
# Rule 133. Graphing the derivative without knowledge of the formulas for dy/dx nor the parent function, is wack:
First, get the slope between two points on the original f(x) and find the slope. As x increases, a decreasing f(x) value indicates a negative slope, while an increasing f(x) value indicates a positive slope. Find an x-value in the middle of the two points and use it as the x-value. Use the slope as the y-value and plug into the new graph, and repeat. Fo a graph with "pointy points", known as corners, they are undefined in the derivative graph (the f'(x)), as they do not have local linearity (see [[[).
# Rule 134. When a fraction has a square root, standard procedure is to multiply it by a conjugate, especially if the square root is on the denominator.
# Rule 135. For the alternate definition of the derivative, you always want to do all the algebra first before substituting for a at the end. Then, you can do a.
# Rule 136. Function f(x) is differentiable on a closed interval [a, b] if it has a derivative at every interior point of the interval, and if the limits below exist at the endpoints.
$$\lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}$$ is the right-hand derivative, and
$$\lim_{h \to 0^-} \frac{f(b + h) - f(b)}{h}$$ is the left-hand derivative.
# Rule 137. The left hand derivative can be found by using the regular f(x), while the right hand derivative is found by using the equation of the f'(x) derivative found from the left hand derivative. If the two derivatives are equal to each other, then the derivative in totality is real.
# Differentiability: The ability to find slope at a point, which is necessary to take the derivative.
# Cases when f'(a) doesn't exist (or, when a point is non-differentiable):
- A Corner: f(x) = |x|
A corner occurs when, at the point a, the function on the graph will make a sharp turn, and it's slope will change instantaneously.
- A Cusp: f(x) = x2/3
A cusp occurs when the line gradually curves into a sharp corner. In effect, the cusp is a special case of the corner: the cusp only has one possible tangent, while a corner has two distinct ones.
- A Vertical Tangent: f(x) = ∛x
When the line is vertical, there is no slope, and thus it is non-differentiable at that point.
- A Discontinuity:
For example, a non-continuous piecewise function. The derivative (nor the limit, if it is a jump discontinuity) exists at a point that is discontinous.
# Local Linearity: If you focus on a point on a curve, if you continuously zoom in on the point, eventually the graph of the curve at the point will resemble a line. Differentiability implies local linearity.
# Rule 138. The symmetric difference quotient is as follows: $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x - h)}{2h}$$ It is used any time you want the derivative from each side of the tangent line (the distance being h), rather than from just one side.
# Rule 139. A derivative is a limit. Not only does the limit have to exist, it must also be continuous. The slope on the left being different from the derivative on the right rules out corners from having derivatives. If the slope becomes a vertical line (e.g., slope/m = infinity), then the derivative also does not exist.
# General Rules of Derivatives:
- Constant Rule: (d/dx) c = 0
- Power Rule: (d/dx) xn = nxn-1
- Trigonometric Rules: (d/dx) sin(x) = cos(x)
(d/dx) cos(x) = -sin(x) - Exponential Rule: (d/dx) bx = bx × ln(b)
- Logarithmic Rule: (d/dx) ln(x) = 1/x
# Rule 140. For a derivative to be, it must have local linearity. No matter how much you zoom into a corner or cusp, they will never have local linearity, and this is why they are considered "non-differentiable" entities.
# Rule 141. When taking the derivative at zero of any function f(x) = xeven/odd, for any matching exponent value, it will never exist. xeven/odd will always produce a corner or cusp at x=0, and thus cannot have a derivative at that point.
# Theorem of Differentiability: Differentiability implies continuity. However, continuity does not imply differentiability. If f has a derivative at x = a, then f is continuous at x = a.
# Intermediate Value Theorem (for Derivatives): If a and b are any two points in an interval on which f is differentiable, the f' takes on every value between f'(a) and f'(b).
# Rule 142. Laws of Differentiation
As used below, all functions f(x), g(x), and h(x), are differentiable functions, and n & c are constants.
- Law 1: Derivative of a Constant.
f(x) = c
c ∈ ℝ
f'(x) = 0.
Examples: f(x) = 5, f'(x) = 0, slope is 0.
- Law 2: Power Rule.
f(x) = xn
f'(x) = n × xn-1
Examples: f(x) = x², f'(x) = 2x.
- Law 3: Constant Multiple Rule.
f(x) = c × h(x)
f'(x) = c × h'(x)
Examples: f'(x) = 2x³, f'(x) = 2[(x³)'], f'(x) = 6x²
f(x) = 4x⁶ + 2x10, f'(x) = 24x⁵ + 20x⁹
- Law 4: Sum and Difference Rule.
(f(x) ± g(x))' = f'(x) ± g'(x)
Examples: f(x) = 2x + x², f'(x) = 2 + 2x
# Rule 143. In order to find the horizontal tangents of a curve, you must find the derivative and then make it equal to 0, and then factor as you see fit.
# Product Rule:
(f × g)' = f'g + fg'
# Quotient Rule:
(f / g)' = (f'g - fg') / (g²)
# Higher Order Derivatives:
- y' = (dy / dx) is the first derivative of y with respect to x.
- y'' = (d²y / dx²) is the second derivative of y - double prime.
- y''' = (d³y / dx³) is the third derivative of y - triple prime.
- y(n) = (dny / dxn) is the nth derivative of y - nth prime.
# Rule 144. Derivatives are allowed to be plugged in over addition and subtraction operations, such as (d/dx)(7v - 3u), requiring the substitution of the derivatives of v and u into the equation.
# Rule 145. The derivative is the slope (rate of change) at any given point on a curve: If you are given a graph of x and asked to find the graph of the "rate of change", just find the derivative graph: The values to be plugged into the Product or Quotient rules are already given!
?. Particle Motion.
# Acceleration: If v(t) represents the velocity of an object, then the acceleration of the object is given by a(t) = v'(t) = s''(t). This represents the rate at which the rate of change is changing (see Rule 147).
# Rule 147. Velocity is the speed in relation to something, so a ball can have negative and positive velocity as it is thrown in the air. Speed is just how fast it's going, and nothing can ever go negative speed.
Acceleration is the derivative of velocity, is the derivative of speed. Acceleration is positive wherever the line of the velocity graph is increasing and negative when decreasing.
# Fundamental Particle Motion Terms:
Displacement: S(tf) - S(ti), in effect (PositionFinal - PositionInitial)
Average Velocity: (∆S / ∆t)
Instantaneous Velocity: (ds/dt), v(t) = s'(t)
# Rule 148. Displacement is the distance between where you started from and where you ended up.
Total distance is how much distance you covered. For example, the displacement can be zero because you can end up exactly where you started.
# Application of Calculus: Cost Modeling:
In manufacturing, the cost of production c(x) represents the cost of producing x number of units. The marginal cost is the rate of change, or "the cost to produce one more item". Marginal cost is the derivative of cost: c'(x).
Revenue, r(x) is the total amount of money collected for the sales of a product. Marginal Reveue is the total amount collected from selling one more unit. r'(x) = Marginal Revenue.
# Sine-Cosine Derivative Loop:
# Rule 149. To find when a particle is moving forward or backward, make a number line for the x's of the velocity graph and find when it is negative or positive: positive is forward, negative is backward.
For where the particle is speeding up or slowing down, if the particle has v(t) > 0 & a(t) < 0, or v(t) < 0 & a(t) > 0 (e.g., the signs do not match between the velocity and acceleration), then the particle is slowing down.
If the particle has v(t) < 0 & a(t) < 0 or v(t) > 0 & a(t) > 0 (matching signs between velocity and acceleration) then the particle is speeding up.
# Rule 150. For equations that have two functions that are just sum or difference and not product nor quotient, you can find the derivative of each individual function and continue the sum and difference as so:
f(x) = sin(x) - cos(x)
f'(x) = cos(x) + sin(x)
# Rule 151. Particle Motion in a Nutshell:
Displacement: The change in position of the particle. S(tf) - S(ti)
Total Distance: The cumulative distance traveled between the starting and ending points of the particle. Found by finding every point in which the velocity changes sign, and absolute valuing all distances traveled under negative velocities while summing everything up.
Speeding up: The v(t) and a(t) of the particle have the same sign.
Slowing down: The v(t) and a(t) of the particle have differing signs.
?. Inverses.
- Differentiate both sides with respect to x.
- Collect the terms with dy/dx on one side of the equation.
- Factor out dy/dx.
- Solve for dy/dx.
# Rule 153. For Implicit Differentiation, only when finding the derivative of y do you need to get dy/dx and use the chain rule. For x, you can just find the derivative with the power rule.
# Rule 154. When writing powers of trigonometric functions, like y = sin³(x), always rewrite the function like y = (sin(x))³. This will make it easier to take derivatives, if needed, and is just a simpler way of seeing the exponential.
# Rule 155. To find the derivative of the inverse of a function, use the formula 1/f'(g(x)) without multiplying by the derivative of the inner function.
# Rule 156. To find the implicit differentiation of trigonometric functions, you just take the derivative of the function itself multiplied by the y': sin(y) = cos(y) × dy/dx.
# Rule 157. The inverse of the function f(x) = y is found simply by reversing the x & y values in the function. If f is differentiable at every point of an interval I, and df/dx is never zero along I, then f has an inverse and f-1 is differentiable at every point on interval f(I).
df/dx is never zero because a function only has an inverse when it is ALWAYS increasing or decreasing. ([[[must past horiz. and vert. line test]]])
df/dx = 0 means a horizontal tangent not increasing or decreasing at that point. Horizontal tangents can denote a change in direction of the function, as long as the sign of df/dx changes from before and after the found horizontal tangent point.
# Inverse Trigonometric Derivatives:
(d/dx) sin⁻¹(x) = (1 / √1 - x²) |x| < 1.
(d/dx) csc⁻¹(x) = (-1 / |x|√x² - 1) |x| > 1.
(d/dx) cos⁻¹(x) = (-1 / √1 - x²) |x| < 1.
(d/dx) sec⁻¹(x) = (1 / |x|√x² - 1) |x| > 1.
(d/dx) tan⁻¹(x) = (1 / 1 + x²) all real #s.
(d/dx) cot⁻¹(x) = (-1 / 1 + x²) all real #s.
# Rule 158. In order to find the derivative of the inverse of a function when only knowing the x-value & the function formula, plug in the x-value into the function to get the y-value, and then do 1/f'(y).
In doing so, you find the derivative of the function while plugging in the y-value under f⁻¹(x) = 1/f'(y).
?. Chain Rule.
(f(g(x)))' = f'(g(x)) × g'(x)
# Integers with Exponential Powers:
For a > 0, and a ≠ 1, (au)' = au × ln(a) × (du/dx)
Without the chain rule for the exponent, this simplifies to (ax)' = ax × ln(a).
# Rule 159. (ex)1 is just ex. Whenever finding the derivative of an equation involving ex, just copy that original ex expression and multiply by the chain rule.
# Rule 160. For other expressions where x is part of the exponent, like 9-x or 3csc(x), you copy the original equation, multiply by ln of the base, and by the derivative of the exponent: 3csc(x) = ln(3) × 3csc(x) × -csc(x)cot(x). See the Integers with Exponential Powers blue section.
# Rule 161. For finding the derivative of an equation with ln, know that (ln(x))' = 1/x, with the x changing as needed. The derivative is just the changed value of 1/x multiplied by the derivative of the x-value: (ln(x²))' = (1/x²) × 2x.
