# Introduction to Differential Calculus

• ### 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) consider the problem of instantaneous rates of the change. They reached independently to the invention of differential calculus. After the development of calculus, mathematics becomes a powerful tool for dealing with rates of change and describing […]

• ### Average and Instantaneous Rate of Change

A variable which can assign any value independently is called independent variable and the variable which depends on some independent variable is called the dependent variable. For Example:             If etc, then We see that as behaves independently, so we call it the independent variable. But the behavior of or depends on the variable . […]

• ### Examples of Average and Instantaneous Rate of Change

Example: Let (a) Find the average rate of change of with respect to over the interval . (b) Find the instantaneous rate of change of with respect to at the point . Solution: (a) For Average Rate of Change:             We have                         Put                         Again Put                         The average rate of […]

• ### Derivative of a Function

Let be a given function of . Given to a small increment and let the corresponding increment of by , so that when becomes , then becomes and we have                                     Dividing both sides by , then                         Taking limit of both sides as             Thus, if be the function of , […]

• ### Examples of Derivative by Definition

Example: Find, by definition, the derivative of function with respect to . Solution: Let I. Change to and to II. Find by subtraction III. Divide both sides by IV. Find the limit of where Which is the derivative of w.r.t . Find, by definition, the derivative of function with respect to . Solution: Let I. […]

• ### Examples of General Theorems

Example: Find if Solution: We have Differentiate w.r.t ‘x’ and using product rule Example: Differentiate with respect to ‘x’. Solution: We have Differentiate w.r.t ‘x’ and using quotient rule

• ### Examples of Trigonometric Differentiation

Example: Differentiate with respect to ‘x’. Solution: Let Differentiate : w.r.t ‘x’

• ### Derivative of Constant Function

The derivative of a constant function is zero. Now we shall prove this constant function with the help of definition of derivative or differentiation. Let us suppose that where is any real constant. First we take the increment or small change in the function. Putting the value of function in the above equation, we get […]

• ### Derivative of x is 1

The derivative of x is 1 (one). Now by definition or first principle we shall show that derivative of x is equal to 1 . Let us suppose that First we take the increment or small change in the function. Putting the value of function in the above equation, we get Dividing both sides by […]

• ### Power Rule of Derivatives

Power Rule of derivative is an essential formula in differential calculus. Now we shall prove this formula by definition or first principle. Let us suppose that the function of the form , where is any constant power. First we take the increment or small change in the function. Putting the value of function in the […]