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# Derivatives of Polynomial Functions

Derivatives are used in Calculus to measure the rate of change of a function with respect to a variable. The use of derivatives is very important in Mathematics. It is used to solve many problems in mathematics like to find out maxima or minima of a function, slope of a function, to tell whether a function is increasing or decreasing. If a function is written as y = f(x) and we want to find the derivative of this function then it will be written as dy/dx and can be pronounced as the rate of change of y with respect to x.

## The derivative of a polynomial function

To calculate the derivative of a polynomial function, first, you should know the product rule of derivatives and the basic rule of the derivative.

### Product rule of derivative

[Tex]\frac{\partial (x^{n})}{\partial x} = n\times x^{n-1}[/Tex]

(Here n can be either positive or negative value)

Understand in this way: The old power of the variable is multiplied with the coefficient of the variable and the new power of the variable is decreased by 1 from the old power.

Example: Find the derivative of x3?

Solution:

Let y = x3

[Tex]=> \frac{\partial y}{\partial x} = 3\times x^{3-1} = 3x^2[/Tex]

### Some basic rules of derivative

• If y = c f(x)

[Tex]\frac{\partial y}{\partial x} = c\frac{\partial (f(x))}{\partial x}[/Tex]

• If y = c

[Tex]\frac{\partial y}{\partial x} = 0[/Tex]

•  [Tex]If \ y= f_{1}(x)\pm  f_{1}(x)[/Tex]

[Tex]\frac{\partial y}{\partial x} = \frac{\partial (f_{1}(x))}{\partial x}\pm \frac{\partial (f_{1}(x))}{\partial x}\\[/Tex]

Example 1: Find the derivative of 4x3 + 7x?

Solution:

Let y = 4x3 + 7x

[Tex]\frac{\partial y}{\partial x} = \frac{\partial (4x^{3})}{\partial x}+\frac{\partial (7x)}{\partial x} \\ \frac{\partial y}{\partial x} = 4\times 3\times x^{2} + 7 = 12x^2 + 7[/Tex]

Example 2: Find the derivative of 3x2 – 7?

Solution:

Let y = 3x2 – 7

[Tex]\frac{\partial y}{\partial x}=6x[/Tex]

## Some more examples on derivative of polynomials

Example 1: Find the derivative of [Tex]\frac{1}{x^{7}}[/Tex]?

Solution:

[Tex]Let \ y=\frac{1}{x^{7}}\\[/Tex]

This can be written as

y = xâˆ’7

[Tex]\frac{\partial y}{\partial x} = (-7)\times x^{-8}[/Tex]

Example 2: Find the derivative of 7x5 + x3 âˆ’ x?

Solution:

Let y = 7x5 + x3 âˆ’ x

[Tex]\frac{\partial y}{\partial x}=35x^{4}+3x^{2}-1[/Tex]

Example 3: Find the derivative of (x + 5)2 + 6x3 âˆ’ 4?

Solution:

Let y = (x + 5)2 + 6x3 âˆ’ 4

[Tex]\frac{\partial y}{\partial x} = 2(x+5)+18x^{2}[/Tex]

Example 4: Find the derivative of 6x3 + (6x + 5)2 âˆ’ 8x?

Solution:

Let y = 6x3 + (6x + 5)2 âˆ’ 8x

[Tex]\frac{\partial y}{\partial x} = 18x^{2}+2(6x+5)(6)-8\\ \frac{\partial y}{\partial x} =18x^{2}+12(6x+5)-8[/Tex]

Example 5: Find the derivative of [Tex]\frac{1}{(2x+8)^{7}}[/Tex]?

Solution:

[Tex]Let \ y=\frac{1}{(2x+8)^{7}}\\ y=(2x+8)^{-7}\\ \frac{\partial y}{\partial x}=(-7)(2x+8)^{-8}(2)\\ \frac{\partial y}{\partial x}=(-14)(2x+8)^{-8}[/Tex]

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