Integration by parts is another technique for simplifying integrands. As we saw in previous posts, each differentiation rule has a corresponding integration rule. In the case of integration by parts, the corresponding differentiation rule is the Product Rule. The technique of integration by parts allows us to simplify integrands of the form: $$ \int f(x) g(x) dx $$

## L'Hospital's Rule for Calculating Limits and Indeterminate Forms

L'Hospital's Rule allows us to simplify the evaluation of limits that involve indeterminate forms. An indeterminate form is defined as a limit that does not give enough information to determine the original limit. The most common indeterminate forms that occur in calculus and other areas of mathematics include:

$$ \frac{0}{0}, \qquad \frac{\infty}{\infty}, \qquad 0 \times \infty, \qquad 1^\infty, \qquad \infty - \infty, \qquad 0^0, \qquad \infty^0 $$

## The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a theorem that connects the two branches of calculus, differential and integral, into a single framework. We saw the computation of antiderivatives previously is the same process as integration; thus we know that differentiation and integration are inverse processes. The Fundamental Theorem of Calculus formalizes this connection. The theorem is given in two parts.

## Indefinite Integrals

As we noted in the previous sections on the Fundamental Theorem of Calculus and Antiderivatives, indefinite integrals are also called antiderivatives and are the same process. Indefinite integrals are expressed without upper and lower limits on the integrand, the notation $\int f(x)$ is used to denote the function as an antiderivative of $F$. Therefore, $\int f(x) \space dx = F^\prime(x)$.

## Substitution Rule

The Substitution Rule is another technique for integrating complex functions and is the corresponding process of integration as the chain rule is to differentiation.

The Substitution Rule is applicable to a wide variety of integrals, but is most performant when the integral in question is of the form:

$$ \int F\big(g(x)\big) g^\prime (x) \space dx $$

## Antiderivatives

Antiderivatives, which are also referred to as indefinite integrals or primitive functions, is essentially the opposite of a derivative (hence the name). More formally, an antiderivative $F$ is a function whose derivative is equivalent to the original function $f$, or stated more concisely: $F^\prime(x) = f(x)$.

The Fundamental Theorem of Calculus defines the relationship between differential and integral calculus. We will see later that an antiderivative can be thought of as a restatement of an indefinite integral. Therefore, the discussion of antiderivatives provides a nice segue from the differential to integral calculus.

## Newton's Method for Finding Equation Roots

Newton's method, also known as Newton-Raphson, is an approach for finding the roots of nonlinear equations and is one of the most common root-finding algorithms due to its relative simplicity and speed. The root of a function is the point at which $f(x) = 0$. Many equations have more than one root. Every real polynomial of odd degree has an odd number of real roots ("Zero of a function," 2016). Newton-Raphson is an iterative method that begins with an initial guess of the root. The method uses the derivative of the function $f'(x)$ as well as the original function $f(x)$, and thus only works when the derivative can be determined.

## Implicit Differentiation

# Implicit Differentiation¶

An explicit function is of the form that should be the most familiar, such as:

$$ f(x) = x^2 + 3 $$ $$ y = \sin{x} $$

Whereas an

*implicit function*defines an algebraic relationship between variables. These functions have a form similar to the following:$$ x^2 + y^2 = 25 $$ $$ y^5 + xy = 3 $$

## The Chain Rule of Differentiation

The chain rule is a powerful and useful derivation technique that allows the derivation of functions that would not be straightforward or possible with the only the previously discussed rules at our disposal. The rule takes advantage of the "compositeness" of a function. For example, consider the function:

## Limit of a Function

A function limit, roughly speaking, describes the behavior of a function around a specific value. Limits play a role in the definition of the derivative and function continuity and are also used in the convergent sequences.

Before getting to the precise definition of a limit, we can investigate limit of a function by plotting it and examining the area around the limit value.

## Derivatives of Logarithmic Functions

Implicit differentiation, which we explored in the last section, can also be employed to find the derivatives of logarithmic functions, which are of the form $y = \log_a{x}$. This also includes the natural logarithmic function $y = \ln{x}$.

## Proving $\frac{d}{dx} (\log_a{x}) = \frac{1}{x \ln{a}}$¶

Taking advantage of the fact that $y = \log_a{x}$ can be rewritten as an exponential equation, $a^y = x$, we can state the derivative of $\log_a{x}$ as:

## Product, Quotient and Power Rules of Differentiation

Several rules exist for finding the derivatives of functions with several components such as $x \space sin \space x$. With these rules and the chain rule, which will be explored later, any derivative of a function can be found (assuming they exist). There are five rules that help simplify the computation of derivatives, of which each will be explored in turn.

## Continuous Functions

# Continuous Functions¶

A function is said to be continuous at a point $a$ if the following statements hold:

- the function $f$ is defined at $a$
- the limit $\lim_{x \to a} \space f(x)$ exists
- the limit is equal to $f(a)$, $\lim_{x \to a} \space f(x) = f(a)$

Continuity of a function can also be expressed more compactly by the statement: $f(x) \to f(a) \space \text{as} \space f \to a$

## Measuring Sensitivity to Derivatives Pricing Changes with the "Greeks" and Python

The Greeks are used as risk measures that represent how sensitive the price of derivatives are to change. This is useful as risks can be treated in isolation and thus allows for tuning in a portfolio to reach a desired level of risk. The values are called 'the Greeks' as they are denoted by Greek letters. Each will be presented in turn as an introduction:

## Black-Scholes Formula and Python Implementation

The Black-Scholes model was first introduced by Fischer Black and Myron Scholes in 1973 in the paper "The Pricing of Options and Corporate Liabilities". Since being published, the model has become a widely used tool by investors and is still regarded as one of the best ways to determine fair prices of options.

## Implied Volatility Calculations with Python

Implied volatility $\sigma_{imp}$ is the volatility value $\sigma$ that makes the Black-Scholes value of the option equal to the traded price of the option.

Recall that in the Black-Scholes model, the volatility parameter $\sigma$ is the only parameter that can't be directly observed. All other parameters can be determined through market data (in the case of the risk-free rate $r$ and dividend yield $q$ and when the option is quoted. This being the case, the volatility parameter is the result of a numerical optimization technique given the Black-Scholes model.

## Put-Call Parity of Vanilla European Options and Python Implementation

In the paper, it is stated the premium of a call option implies a certain fair price for the corresponding put option (same asset, strike price and expiration date). The Put-Call Parity is used to validate option pricing models as any pricing model that produces option prices which violate the parity should be considered flawed.

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