## Introduction to Functions

In mathematics, the term function is very famous. If we look at the historical background the term, function was first used… Click here to read more

From basic to higher mathematics

In mathematics, the term function is very famous. If we look at the historical background the term, function was first used… Click here to read more

Let $$f\left( x \right)$$ be a real valued function if the value of the function $$f\left( x \right)$$ approaches a… Click here to read more

If $$f$$ is a real valued function, then $$x$$ can approach $$a$$ from two sides: the left side of $$a$$… Click here to read more

In this tutorial we give the statements of theorems on limits which will be useful in evaluating limits. (1) The… Click here to read more

In this tutorial we shall look at the limit of a polynomial function of any degree, and this is useful… Click here to read more

In this tutorial we shall derive the limit of a radical expression. These types of limits are usually solved by… Click here to read more

In this tutorial we shall discuss the limit of a piecewise function. Let us consider an example of the limit… Click here to read more

So far we have discussed limits at some fixed numbers. In this section we shall be concerned with limits at… Click here to read more

So far we have discussed limits at some fixed numbers. In this section we shall be concerned with the limits… Click here to read more

In this tutorial we shall discuss an example related to the limit of a function at positive infinity, i.e. $$x… Click here to read more

In this tutorial we shall discuss an example related to the limit of a function at negative infinity, i.e. $$x… Click here to read more

In this tutorial we shall discuss an example related to the limit at negative infinity with the radial form of… Click here to read more

In this tutorial we shall discuss an example related to the limit at positive infinity with the radial form of… Click here to read more

In this tutorial we shall discuss the very important formula of limits, \[\mathop {\lim }\limits_{x \to \infty } {\left( {1… Click here to read more

In this tutorial we shall discuss another very important formula of limits, \[\mathop {\lim }\limits_{x \to 0} \frac{{{a^x} – 1}}{x}… Click here to read more

In this tutorial we shall discuss an example of limit which involves quadratic functions, and to find the value of… Click here to read more

In this tutorial we shall discuss an example of evaluating limits involving radical expressions. In most cases, if the limit… Click here to read more

In this tutorial we shall discuss an example of evaluating limits involving cubic expression. In most cases, if a limit… Click here to read more

In this tutorial we shall discuss an example of evaluating limits involving a function with nth power of variable. In… Click here to read more

A moving physical object cannot vanish at one point and reappear someplace else to continue its motion. Thus, we perceive… Click here to read more

Let’s take an example to find the continuity of a function at any given point. Consider the function of the… Click here to read more

In the seventeenth century, Sir Isaac Newton, an English mathematician (1642–1727), and Gottfried Wilhelm Leibniz, a German mathematician (1646–1716), considered… Click here to read more

A variable which can assign any value independently is called the independent variable, and the variable which depends on some independent… Click here to read more

Example: Let $$y = {x^2} – 2$$ (a) Find the average rate of change of $$y$$ with respect to $$x$$… Click here to read more

Let $$y = f(x)$$ be a given function of $$x$$. Give to $$x$$ a small increment $$\Delta x$$ and let… Click here to read more

Example: Find, by definition, the derivative of function $${x^2} – 1$$ with respect to $$x$$. Solution: Let \[y = {x^2}… Click here to read more

The derivative of a constant function is zero. Now we shall prove this constant function with the help of the… Click here to read more

The derivative of x is 1 (one). Now by the definition or first principle we shall show that the derivative… Click here to read more

The Power Rule of derivatives is an essential formula in differential calculus. Now we shall prove this formula by definition… Click here to read more

It is given that the derivative of a function that is the sum of two other functions, is equal to… Click here to read more

It is given that the derivative of a function that is the difference of two other functions, is equal to… Click here to read more

To find a derivative of a function, consider the following examples to expand on the concept of a derivative of… Click here to read more

Finding a derivative of the square roots of a function can be done by using derivative by definition or the… Click here to read more

The product rule of derivatives is $$\frac{d}{{dx}}\left[ {f\left( x \right)g\left( x \right)} \right] = f’\left( x \right)g\left( x \right) +… Click here to read more

The quotient rule of derivatives is $$\frac{d}{{dx}}\left[ {\frac{{f\left( x \right)}}{{g\left( x \right)}}} \right] = \frac{{g\left( x \right)f’\left( x \right) –… Click here to read more

Composition of Two Functions The composition $$f \circ g$$ of two functions $$f$$ and $$g$$ is defined as \[\left( {f… Click here to read more

Example 1: Differentiate $$y = {\left( {2{x^3} – 5{x^2} + 4} \right)^5}$$ with respect to $$x$$ using the chain rule… Click here to read more

In this tutorial we will discuss the basic formulas of differentiation for algebraic functions. 1. $$\frac{d}{{dx}}\left( c \right) = 0$$,… Click here to read more

We shall prove the formula for the derivative of the sine function by using definition or the first principle method…. Click here to read more

We shall prove the formula for the derivative of the cosine function by using definition or the first principle method…. Click here to read more

We shall prove the formula for the derivative of the tangent function by using definition or the first principle method…. Click here to read more

We shall prove the formula for the derivative of the cotangent function by using definition or the first principle method…. Click here to read more

We shall prove the formula for the derivative of the secant function using definition or the first principle method. Let… Click here to read more

We shall prove the formula for the derivative of the cosecant function by using definition or the first principle method…. Click here to read more

In this tutorial we shall discuss the derivative of the sine squared function and its related examples. It can be… Click here to read more

In this tutorial we shall discuss the derivative of the cosine squared function and its related examples. It can be… Click here to read more

In this tutorial we shall discuss the derivative of the tangent squared function and its related examples. It can be… Click here to read more

In this tutorial we shall discuss the derivative of the cotangent squared function and its related examples. It can be… Click here to read more

In this tutorial we shall discuss the derivative of the secant squared function and its related examples. It can be… Click here to read more

In this tutorial we shall discuss the derivative of the cosecant squared function and its related examples. It can be… Click here to read more