Part 1 Differential Calculus

Definition and Basics

The differential calculus (or differentiation) is an essential part of analysis and therefore a field of mathematics. The central topic of differential calculus is the calculation of local changes of functions.

While a continuous function continuously assigns certain output values to its input values, differential calculus determines how strongly the output values change in response to very small changes in the input values. It is closely related to integral calculus, with which it is collectively referred to as infinitesimal calculus.

The derivative of a function is used to represent local changes in a function and is at the same time a fundamental concept of differential calculus. Instead of derivative, the term diff

erential quotient is also used; its geometric interpretation is the slope of the tangent line.

Part 2 Differentiation Rules

Differentiation Rules (or Derivative Rules)

Differentiation rules (or derivative rules) are mathematical calculation rules used in differential calculus. They make it possible to determine the derivative (the local rate of change or slope) of functions without having to recalculate the general limit (the difference quotient) every time.

An Overview of the Most Important Differentiation Rules

• Constant Factor Rule: A constant factor remains unchanged when taking the derivative.
• Sum and Difference Rule: Sums or differences of functions are differentiated by differentiating the individual terms separately and then adding or subtracting the results, respectively.
• Power Rule: A power of a power function is differentiated by bringing the exponent to the front and reducing the exponent by 1.
• Product Rule: Used when two functions are multiplied together.
• Quotient Rule: Used when one function is divided by another function.
• Chain Rule: Used for functions nested within one another (derivative of the outer function × derivative of the inner function).

Part 3 Curve Discussion

Graphical Differentiation - Curve Discussion

Graphical differentiation is a method in which the graph of a derivative function f′(x) is drawn in a coordinate system solely by observing and analyzing the behavior of a given original function f(x). We will define a complete curve discussion with all possible results

Definition

When differentiating graphically (also called graphical differentiation), the slope of a function at different points is determined by visual inspection using a set square and ruler instead of calculating it with a formula. Marked points on the original curve are used for this purpose.

• Extrema (local maxima and minima):
   
• Slope behavior (monotonicity):
• Points of inflection:
  Here, the slope has its steepest value (an extreme value), which leads to a local extremum (maximum or minimum) in the derivative graph.
• Estimating the slope
• Tangent to a Function

Part 4 Limit Calculation Using l'Hospital 

L’Hospital’s Rule

L’Hospital’s Rule  (pronounced [lopi’tal]) can be used to calculate limits of quotients of two functions that either converge to zero or diverge in a determinate manner, using the first derivatives of these functions. An analogous statement for sequences instead of functions is the Stolz–Cesàro theorem.

The rule is named after Guillaume François Antoine, Marquis de L’Hospital (1661–1704).

L’Hospital published it in 1696 in his book.

Part 5 Partial Differentiation

Definition and Applications of Partial Differentiation

A partial derivative is the derivative of a function with several variables with respect to only one of these variables, while all other variables are treated as constants.

Definition

If a function  has several independent variables, it is differentiated partially with respect to a variable of interest while keeping all the other variables fixed.

Practical procedure: When differentiating, all other variables behave like ordinary numbers (constants). The familiar one-dimensional differentiation rules, such as the product rule and chain rule, apply.

Applications

Partial derivatives are used in many areas of the natural sciences, economics, and engineering:

• Finding extrema: Searching for maxima and minima—for example, of cost or profit functions with several parameters—is done by setting all partial derivatives equal to zero (gradient = 0).
• Physics and engineering mechanics: Description of fields, such as temperature or fluid-flow distributions, using gradient, divergence, and curl.
• Partial differential equations: Modeling physical laws that depend both on position and on time, such as the heat equation or wave equation.
• Error and uncertainty analysis: Calculation of the total differential to estimate measurement errors when a result depends on several error-prone input variables.