**Simple Definition of Derivative in Calculus**

Derivative is the rate of change of a function f(x) with respect to any variable.

**Derivative**hide

**Derivative of a Function Examples**

Let **f **be a real valued function Continuous in the interval (**x**,**x _{1}**)

**⊆**

**D**the domain of (

_{f}**f**) , then

different quotient=

**f(x**

_{1}) – f(x)/x_{1}– xrepresents the average rate of change in the value of f with respect to change

**x**in the values of independent variable x.

_{1}– x### Uses of Derivatives in Economics

- we mostly use derivative to determine minimum and maximum values of a particular function f(x).
- Derivatives are also used in many engineering and science problems.
- Specially modelling the behavior of any moving object.

**How Can We Use Derivatives in our Daily Life?**

- ⇒ Derivative is mostly used in calculating profit and loss in business by using different graphs.
- ⇒ it is also used to determine speed or any distance covered i.e, miles per hour or kilometer(km) per hour.
- ⇒ Derivatives are most commonly used in physics to drive different equations.

**How to Find Derivative of any Function f(x)?**

Basically it is obtained by moving the power in the start and decreasing the power by **1**.

**Example for Derivative of any Function f(x) **

Derivative of **x⁴** is **4x³**. Note ⇒ that power is in the start and power is decreased by** 1**

Derivative of **x**^{12} is **12x**^{12-1} **= 12x**^{11}

**Some more examples for Derivative of Function **

\begin{array}{l}4x^3+\;x^2\;\\=\;\frac d{dx}\left(4x\right)^3\;+\;\;\frac d{dx}\left(x\right)^2\\=3\left(4x\right)^{3-1}\;+\;2\left(x\right)^{2-1}\\=12x^2\;+\;2x\\or\;2x(6x\;+\;1)\end{array}

### Derivative Notation and Mathematician Names

Now, we write in the table the notations for the derivatives of y=f(x) used by different mathematicians.

Name of Mathematician | Leibniz | Newton | Lagrange | Cauchy |

Notation Used For Derivative | dy / dx or df / dx | f(x) | f’(x) | D f(x) |

**Basic Rules of Differentiation with Examples**

⇒ Derivative of a constant is always** zero** i.e., derivative of (**x² + 1**) is

\begin{array}{l}x^2+\;1\;\\=\;\frac d{dx}\left(x\right)^2\;+\;\;\frac d{dx}1\\=2\left(x\right)^{2-1}\;+\;0\\=2x\\\end{array}

⇒Derivative of **d/dx(x) =1**

In mathematics **a,b,c** are mostly considered constants. So, derivative of a,b,c are also** zero(0)**.

Derivative of a sum or a difference of any function

in simple manners:

if f and g are differentiable at x then,**[f(x) + g(x)]’= f’(x) – g’(x)**that is,

**=d/dx[f(x) + g(x)]**

= d/dx[f(x) + d/dx [g(x)]

= d/dx[f(x) + d/dx [g(x)]

also,**=[f(x) – g(x)]’= f’(x) – g’(x)**that is,

**=d/dx [f(x) – g(x)]**

=d/dx [f(x)] -d/dx[g(x)]

=d/dx [f(x)] -d/dx[g(x)]

## What is the Product and Quotient Rule with Example

Product and Quotient rule with formula are as follow;

**What is Product Rule in Differentiation?**

if **f** and **g** are differentiable at x , then **fg** is also differentiable at **x** and**[f(x)g(x)]’ = f’(x) g(x) + f(x) g’**

*that is,*

**d/dx[f(x) g(x)] = [d/dx[f(x)] g(x) + f(x) [d/dx[g(x)]**

**What is Quotient Rule in Differentiation**

if **f** and **g** are differentiable at **x** and **g≠0**,

for any **x∈ D(g)** then **f/g** is differentiable at **x** and **[f(x)/g(x)]’ = f’(x) g(x) – f(x) g’(x)/[g(x)]²**

*that is,*

**d/dx[f(x)/g(x)]’****= [d/dx[f(x)] g(x) – f(x)[d/dx[g(x)]/[g(x)]²**