Partial Derivative - the partial derivative of f(x) with several variables (x, y) is the derivative with respect to one of those variables while the others are held as constant

• the partial derivative is written as ∂f/∂x (∂ is called curly d or “del”) or fx or fy (x/y is written in subscript, so ‘f sub x’ or ‘f sub y’)

• If z = f(x, y), then ∂z/∂x is asking for the partial derivative with respect to ‘x’ (and keep ‘y’ as constant) and ∂z/∂y is asking for the partial derivative with respect to ‘y’ (and keep ‘x’ as constant)

• Same rule applies for three variables; e.g. when finding partial derivative for f(x, y, z), f(x) differentiate x while keeping y and z as constants

Rules when finding partial derivatives:

Chain Rule Example: z = x^3y + 3x^2 + y^3, x = 4 + t^2, y = 5t^3

Composite functions. Must use the chain rule to find dz/dt. The formula is [∂z/∂x * dx/dt] + [∂z/∂y * dy/dt]

1. Partial derivative of z with respect to x and keep y constant

2. Differentiate x with respect to t

3. Multiply results from 1 and 2

4. Partial derivative of z with respect to y and keep x constant

5. Differentiate y with respect to t

6. Multiply results from 4 and 5

7. Add results from 3 and 6

Higher Order Partial Derivatives - or “mixed partial derivative”; in the case of 2 (or more) variables, the partial derivative is taken with respect to one variable. Then, the partial derivative is taken from the result (double derivative) but with respect to the other variable. Example: ∂f/∂x (or fx) is the partial derivative of f with respect to x, and ∂/∂y(∂f/∂x) (or fxy) is the partial derivative of the partial derivative of f with respect to y

Clairaut's Theorem - symmetry of second derivatives; this theorem tells us that the mixed variable partial derivatives are equal
▸ fxy(a,b) = fyx(a,b)
▸fxyz = fyxz = fzxy (each variable was differentiated once)
▸fyyx = fyxy = fxyy (x was differentiated once; y was differentiated twice)

Example: f = x^2y^3 + 4xy^2

• fx = 2xy^3 + 4y^2

• fxy = 6xy^2 + 8y

• fy = 3x^2y^2 + 8xy

• fyx = 6xy^2 + 8y

Gradient (∇) - like the derivative, the gradient measures how steep a slope is; it denotes the direction of the greatest change of a scalar function

• While a multi-dimensional function f(x, y, z) has partial derivatives, the gradient of a function ∇f(x) is the collection of all its partial derivatives into a vector; i.e. a full derivative of a multivariable function

• ∇ is an upside-down delta sign called “del”, indicating change; or also called “nabla”; nabla refers to the symbol itself, and del refers to the operator it represents

• Gradient points to the direction of the greatest increase or steepest ascent; not to be confused with coordinates which indicate location and not direction; e.g. if ∇f(x, y, z) = (5, 10, 81), then you want to move towards z which at that point the gradient equals zero (i.e. maxima)

Divergence - measure of the quantity of flux (density) emanating from any point of the vector field

• Formula is ∇ ⋅ F = ∂F/∂x + ∂F/∂y

• Same symbol as Gradient plus dot ∇ ⋅

• Divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point

Zero-divergence - when ∇F is zero, it means the density stays constant (i.e. fluid flows freely)

Sink - negative divergence; more dense after a momentary fluid motion

Source - positive divergence; less dense during a momentary motion

Hat symbol (^) over i and j indicate “unit vector in the direction of i (or j)”. So in this case, it was asking (x^2 - y^2) in the x-direction and 2xy in the y-direction.

Curl - measures the “rotation” in the vector field

• Formula is ∇xF = ∂F(2)/∂x - ∂F(1)/∂y

• Same symbol as Gradient plus x ∇ x

• Positive number suggests counter-clockwise direction, while negative number suggests clockwise direction

• Typically used in three-dimensional fields

Why add “C”? When initially differentiating a function, the constant disappears because the derivative of any constant is zero. Since integration works backward and one does not know if there was a constant involved initially, this “C” is added as a placeholder.

Integration by Parts - (or partial integration) the process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative.

• The formula is ∫u v dx = u∫v dx −∫u' (∫v dx) dx

• Or (shortened) ∫u dv = uv - ∫v du

• When determining which function to assign to ‘u’ go in this order:

  1. inverse trigonometric functions (sin^-1(x))

  2. logarithmic functions (ln(x))

  3. algebraic functions (x^3)

  4. trigonometric functions (cos(x))

  5. exponential functions (e^x)

Limit - the value that a function approaches as the input approaches some value; describes how a function behaves near a point (instead of at that point exactly)

L'Hospital's Rule - if the result is an indeterminate form (e.g., 0/0 or ∞/∞), then this rule tells us to differentiate the numerator and differentiate the denominator and then take the limit; lim x→c f(x)/g(x) = lim x→c f’(x)/g’(x)

Example: f(x) = x^3

1. Derivative ▸ f’(x) = 3x^2 or f’’(x) = 6x

2. Slope of tangent line at x = 2▸mtan = f(2) = 12

3. Slope of secant line at [1, 3] ▸ msec = (y2 - y1) / (x2 - x1) = (f(3)-f(1)) / (3 - 1) = 13

4. Slope of secant line at [1.9, 2.1] ▸ msec = 12.01

5. lim x→2 ((fx) - f(2)) / (x - 2)
   lim x→2 (x^3 - 8) / (x - 2) (factor difference of cubes)
   lim x→2((x - 2)(x^2 + 2x + 4))/(x - 2) (cancel out the (x - 2))
   lim x→2 (x^2 + 2x + 4) = 12

Maxima / Minima
Where the slope of the tangent line to the function is zero. Local maxima/minima references to a particular interval, while absolute maxima/minima is the highest/lowest point of the entire domain. There can be multiple points for local but only one absolute.

Steps to find the local min/max:

1. Set the derivative equal to zero and solve for x or the critical value(s)

2. Plug in an x value that’s higher and lower into f’(x) to figure out the direction of the slope (negative to positive means minimum, positive to negative means maximum).

3. If the slope is zero at the critical point x = c (f’(x)=0), then find the second derivative f’’(x).

4. Plug in the min/max x value into the original function f(x) to find the y-value

5. Final answer would be that the local min/max is located at (x, y)

Factorial - denoted by (!); the product of all positive integers less than or equal to n; exception for zero 0! = 1; formula is z! = z(z - 1)(z - 2)…1

Gamma Function (Γ) - also “Euler's integral of second kind”; the analytic continuation (a technique to extend the domain of definition of a given analytic function) of the factorial defined at Γ(n)=(n−1)! or Γ(z+1)=zΓ(z) - see proof of integration by parts on the right
But why extend? Factorials can only calculate non-negative integers (a discrete set), and the Gamma function helps generalize the factorial into more complex numbers (a continuous set)

• Formula is Γ(z) = lim 0→∞ ∫x^(z-1)e^-x dx

• Or also Γ(z+1) = lim 0→∞ ∫(x^z)(e^-x) dx

• When z is a real number Γ(z+1) = z! or Γ(z) = (z - 1)!

• Defined for all complex numbers except for non-positive integers; accepted inputs are 3, 4.5, -7.2

• Interpolates the factorial function to non-integer values; i.e., it fills in the gaps between the factorial points

• useful for defining several probability distributions, such as Gamma distribution, Chi-squared distribution, Student's t-distribution, and Beta distribution; these distributions are used for Hypothesis Testing, Bayesian Analysis, building statistical models like LDA (Latent Dirichlet Allocation), and in stochastic processes.

• useful for modeling situations involving continuous change

• useful in identities and proofs in analytic contexts

Beta Function (β) - also “Euler's integral of the first kind”; forms an association between the input and output sets in integral equations and many more Mathematical operations

• Formula is β(x, y) = lim 0→1 ∫(t^(x - 1))*((1 - t)^(y - 1)) dt

• Expressed as β(x, y) where x and y are real numbers greater than zero

• Symmetric function β(x, y) = β(y, x)

• Gamma is a one-variable function and Beta is a two-variable function

• Relationship to Gamma: β(x, y) = Γ(x)Γ(y) / Γ(x + y)

• useful for modeling portfolio management through the preferential attachment process

• a component of beta distribution, which is a dynamic, continuously updated probability distribution with two parameters; useful for machine learning when modeling uncertainty about the probability of success of a given experiment