## Introduction to Differential Calculus

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

From basic to higher mathematics

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

In trigonometric differentiation, most of the examples are based on the sine square roots function. We will discuss the derivative… Click here to read more

In this tutorial we will discuss the basic formulas of differentiation for trigonometric functions. 1. $$\frac{d}{{dx}}\sin x = \cos x$$… Click here to read more

In this tutorial we shall discuss the derivative of inverse trigonometric functions and first we shall prove the sine inverse… Click here to read more

In this tutorial we shall discuss the derivative of inverse trigonometric functions and first we shall prove the cosine inverse… Click here to read more

In this tutorial we shall explore the derivative of inverse trigonometric functions and we shall prove the derivative of tangent… Click here to read more

In this tutorial we shall explore the derivative of inverse trigonometric functions and we shall prove the derivative of cotangent… Click here to read more

In this tutorial we shall explore the derivative of inverse trigonometric functions and we shall prove the derivative of secant… Click here to read more

In this tutorial we shall explore the derivative of inverse trigonometric functions and we shall prove the derivative of cosecant… Click here to read more

In this tutorial we shall discuss the basic formulas of differentiation for inverse trigonometric functions. 1. $$\frac{d}{{dx}}{\sin ^{ –… Click here to read more

A function defined by $$y = {\log _a}x,\,\,\,x > 0$$, where $$x = {a^y},\,\,\,a > 0$$, $$a \ne 1$$ is… Click here to read more

If $$y = f\left( x \right)$$ is a complicated function, i.e. it involves several products of functions, quotients or radical… Click here to read more

In this tutorial we shall discuss the basic differentiation formulas of logarithmic functions. 1. $$\frac{d}{{dx}}\ln x = \frac{1}{x},\,\,\,x >… Click here to read more

Example: Differentiate $${\log _{10}}\left( {\frac{{x + 1}}{x}} \right)$$ with respect to $$x$$. Consider the function \[y = {\log _{10}}\left( {\frac{{x… Click here to read more

In this tutorial we shall find the general rules of derivative of exponential functions, and we shall prove the general… Click here to read more

In this tutorial we shall find the derivative of exponential function $${e^x}$$ and we shall prove the general rules for… Click here to read more

In this tutorial we discuss the basic differentiation formulas of exponential functions. 1. $$\frac{d}{{dx}}{a^x} = \frac{1}{{x\ln a}},\,\,\,a > 0,\,\,\,a \ne… Click here to read more

Example: Differentiate $${a^{\sin x}} + {e^{\cos x}}$$ with respect to $$x$$. We have the given function \[y = {a^{\sin x}}… Click here to read more

In this tutorial we shall study certain combinations of $${e^x}$$ and $${e^{ – x}}$$, which are called hyperbolic functions. These… Click here to read more

In this tutorial we shall prove the derivative of the hyperbolic sine function. Let the function be of the form… Click here to read more

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