Table of Contents
Notation of Differentiation
The derivative of a function is a measure of how the function changes as its input changes. The derivative is usually written as a symbol, such as or . The derivative of a function at a point is the slope of the function at that point.
What is Differentiation in Maths?
Differentiation is a process in mathematics that allows you to find the derivative of a function. The derivative is a measure of how a function changes as you move closer to a specific point.
Rules for Important Differentiation Formulas
There are a few key rules to remember when differentiating important formulas.
The first is that the derivative of a constant is always zero. For instance, the derivative of x is zero, and the derivative of y is zero.
The second rule is that the derivative of a sum is the sum of the derivatives. For instance, the derivative of x + y is the sum of the derivatives of x and y.
The third rule is that the derivative of a product is the product of the derivatives. For instance, the derivative of x*y is the product of the derivatives of x and y.
Rules for Compound Functions
A compound function is a function that is composed of two or more functions. The rules for working with compound functions are:
1. The functions in a compound function must be compatible. This means that they must have the same number of inputs and the same number of outputs.
2. The functions in a compound function must be nested. This means that the output of one function must be the input to the next function.
3. The order of the functions in a compound function is important. This means that the output of the first function must be the input to the second function, and so on.
4. The order of the inputs to a compound function is important. This means that the input to the first function must be the input to the second function, and so on.
5. The order of the outputs of a compound function is not important. This means that the output of the last function can be the output of the compound function, or it can be passed on to another function.
Sum or Difference
The sum of two numbers is the result of adding the numbers together. The difference of two numbers is the result of subtracting the numbers.
Product Rule
If \(f(x)\) and \(g(x)\) are both functions, then the derivative of the function \(f(x)\) with respect to \(x\) is the derivative of the function \(g(x)\) multiplied by the coefficient of \(x\) in the function \(f(x)\).
Quotient Rule
The quotient rule states that the derivative of the quotient of two functions is the derivative of the first function divided by the derivative of the second function.
The derivative of a quotient is:
Chain Rule
The Chain Rule states that if is a function of two variables and is a function of then is a function of .
Applications of Differentiation in Real Life Problems
There are countless real life applications of differentiation in a plethora of fields. Some examples include:
Calculating the gradient of a curve in order to find the quickest way to get from one point to another.
Determining the maximum or minimum point of a curve.
Finding the points of inflection on a curve.
Designing products that can be manufactured efficiently and economically.
Optimizing business strategies to ensure the most profit is made.
Forecasting future trends in various markets.
Planning and executing scientific experiments.
There are undoubtedly countless other examples of how differentiation is used in real life, in fields that span from the very mathematical to the very creative. Differentiation is a powerful tool that can be applied in many different ways to help us understand and solve problems.
