Derivative Examples

Derivative Examples

Education

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In mathematics, we use derivatives to show the rate of change. Consider a function that involves the pressure of a gas. The change in pressure is dependent on temperature and volume. As far as the JEE exam is concerned, differentiation is an important topic. The theory of derivative is derived from limits. The method of finding the derivative is known as differentiation.

A function that shows the rate of change of the other function can be called the derivative of that function. By differentiating a function, we can find the derivative. We denote derivative by dy/dx. I.e., the change in y with respect to x. If f(x) is a function, the derivative is represented as f’(x). In this article, we will discuss important rules, formulas, and derivative examples.

The slope of a function represents the derivative of a function.

Slope = change in y / change in x = dy/dx.

Important Rules

  • Constant rule: (d/dx)C = 0
  • Constant multiple rule: (d/dx) C.u = C (du/dx)
  • Power rule: (d/dx)xn = nxn-1
  • Product rule:(d/dx) (u.v) = u.(dv/dx)+v.(du/dx)
  • Sum or difference rule: (d/dx) (u±v) = (du/dx) ± (dv/dx)
  • Quotient rule: (d/dx) (u/v) = [v. (du/dx) – u.(dv/dx))/v2]

Some Basic Derivaties

  1. (d/dx) x = 1
  2. (d/dx) √x = 1/2√x
  3. (d/dx) ex = ex
  4. (d/dx) log x= 1/x
  5. (d/dx) sin x = cos x
  6. (d/dx) cos x = -sin x
  7. (d/dx) tan x = sec2 x

Let us see some examples of derivatives. 

Example 1:

Find the derivative of 2x4 + x2 + 5.

Solution:

Using the power rule, (d/dx) 2x4 = 8x3

(d/dx) x2 = 2x

By constant rule, (d/dx) 5 = 0

So the derivative of 2x4 + x2 + 5 = 8x3 + 2x = 2x(4x2 + 1).

Example 2:

Find the derivative of 5x + x2 sin x

Solution:

(d/dx)5x = 5

(d/dx) x3sin x = 3x2 sin x + x3 cos x

So the derivative  5x + x2 sin x = 5 + 3x2 sin x + x3 cos x

Differential Equation

An equation with a function and one or more of its derivatives is called a differential equation. The function may represent physical quantities and the derivative represents the rate of change. A relationship between a function and a derivative is shown in a differential equation. A function is a relation from a set of inputs to a set of possible outputs, where each input is related exactly to only one output.

Example of a differential equation is y + (dy/dx) =  7 x.

In the above example, y is the function and dy/dx is the derivative of y.

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In the real-world, the population growth rate can be considered as an example. Let y be the population at any time t. Let r be the growth rate, then the population rate of change is dy/dt.The rate of change of population with time is the growth rate r times the population.

So dy/dt = r×y. This is a differential equation because it has a function y(t) and its derivative.

As far as the JEE exam is concerned, differential equations are an important topic. Students are recommended to practise Differential Equations JEE Previous year solved Questions so that they can score high ranks in the examination. Solving previous year questions will help the students to improve their speed of answering the questions.

Students can expect one 4 mark question from the topic differential equation for the JEE main examination. Practising previous year question papers will help students to understand the pattern of the questions asked for the JEE exam. It also improves the confidence of the students.

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