Skip to content
Related Articles

Related Articles

Improve Article
Save Article
Like Article

Derivatives of Polynomial Functions

  • Last Updated : 15 Dec, 2020

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.

Attention reader! All those who say programming isn't for kids, just haven't met the right mentors yet. Join the  Demo Class for First Step to Coding Coursespecifically designed for students of class 8 to 12. 

The students will get to learn more about the world of programming in these free classes which will definitely help them in making a wise career choice in the future.

Product rule of derivative

\frac{\partial (x^{n})}{\partial x} = n\times x^{n-1}



(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

=> \frac{\partial y}{\partial x} = 3\times x^{3-1} = 3x^2

Some basic rules of derivative

  • If y = c f(x)

\frac{\partial y}{\partial x} = c\frac{\partial (f(x))}{\partial x}

  • If y = c

\frac{\partial y}{\partial x} = 0



  •  If \ y= f_{1}(x)\pm  f_{1}(x)

\frac{\partial y}{\partial x} = \frac{\partial (f_{1}(x))}{\partial x}\pm \frac{\partial (f_{1}(x))}{\partial x}\\

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

Solution:

Let y = 4x3 + 7x

\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

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

Solution:

Let y = 3x2 – 7

\frac{\partial y}{\partial x}=6x

Some more examples on derivative of polynomials

Example 1: Find the derivative of \frac{1}{x^{7}}?



Solution:

Let \ y=\frac{1}{x^{7}}\\

This can be written as 

y = x−7

\frac{\partial y}{\partial x} = (-7)\times x^{-8}

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

Solution:

Let y = 7x5 + x3 − x

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

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



Solution:

Let y = (x + 5)2 + 6x3 − 4

\frac{\partial y}{\partial x} = 2(x+5)+18x^{2}

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

Solution:

Let y = 6x3 + (6x + 5)2 − 8x

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

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

Solution:

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}




My Personal Notes arrow_drop_up
Recommended Articles
Page :