Derivatives: Overview, Questions, Preparation

A derivative is the branch of calculus. It is termed as the rate of change of a quantity or a function with respect to other functions or quantity. Calculating the derivative of functions is known as differentiation. The formula for derivative is:
f'(x) = lim_Δx→0{f(Δx +x)−f(x)}/ Δx

If we have a function p(x), then the symbol of its derivative will be p'(x).

If a(x) and b(x) are the functions, then the properties of derivatives will be:

d [a(x) + b(x)] /dx = d(a(x))/dx + d(b(x))/dx

d [a(x) - b(x)] /dx = d(a(x))/dx - d(b(x))/dx

d[a(x) * b(x)] /dx = d [(a(x)* b(x) + a(x)*d(b(x))/dx ] /dx 

d/dx [a(x)/b(x)] = {d/dx [a(x)b(x) - a(x)d(b(x))/dx ]} / (b(x))²

Types of Derivatives

There are generally 2 types of derivatives:

First-Order Derivative: It explains the function's direction, whether it is increasing or decreasing. 

Second-Order Derivative: This derivative helps in providing the graph's shape for the respective function.

Important Applications Derivatives in Mathematics

Derivatives have various applications in mathematics like:

• Normal and Tangent to a curve
• Newton's Method
• Linear Approximations
• Maximum and Minimum Values
• Increasing and Decreasing Functions
• General Formulas of Derivative:

Some of the general formulas are - 

The topic of derivatives has a weightage of nearly 4 to 6 marks. Around 2 to 3 questions are always asked in class 11^th exams, and the questions range from easy to moderate level. The questions of class 12^th also use the concept and applications of Limits and Derivatives. The chapter Limits and Derivatives, and its applications are present in both class 11^th as well as class 12^th. Limits is also a topic from the same chapter; Limit is termed as a value from a function f(x) when f(x) reaches the limit. It is used to define integral calculus, continuity, and

integration of the function. Other topics that are involved in the chapter Limits and Derivatives are:

• Defining Derivatives of a function
• Trigonometric Functions Limits
• Limits and Derivatives

Q: What is Partial Derivative?

Q: What will be the derivative of a constant?

Q: Mention one example of Derivatives that exist in the real world.

Q: What will be the derivative of sinx*cosx?

Q: What is the formula for the limit of a function p(x)?