Derivatives of Polynomial Functions

Derivatives are used in Calculus to measure the rate of change of a function with respect to a variable. They are used to solve many problems in mathematics, like as finding out maxima or minima of a function, determining the slope of a function, or determining 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. 

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

Product Rule

(Here n can be either positive or negative value)

Understand in this way: The old power of the variable is multiplied by 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

Some Basic Rules

• If y = c f(x)

• If y = c

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

Solution:

Let y = 4x³ + 7x

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

Solution:

Let y = 3x² - 7

Some More Examples on Derivative of Polynomials

Example 1: Find the derivative of ?

Solution:

This can be written as 

y = x⁻⁷

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

Solution:

Let y = 7x⁵ + x³ − x

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

Solution:

Let y = (x + 5)² + 6x³ − 4

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

Solution:

Let y = 6x³ + (6x + 5)² − 8x

Example 5: Find the derivative of ?

Solution:

Practice Questions

Question 1: Find the derivative of x³ + 4x² − 6x + 8

Question 2: Find the derivative of 5x⁴ + 3x² − 2x + 1.

Question 3: Find the derivative of 2x⁶ − x⁴ + 3x².

Question 4: Find the derivative of 8x⁷ − 5x³ + 2x.

Question 5: Find the derivative of 9x⁵ + 2x³ − x + 4.

Question 6: Find the derivative of 2/x⁵.

Question 7: Find the derivative of 6x⁴ + x² + 1/x³.

Question 8: Find the derivative of (x+2)³ + 4x⁵ − x².

Question 9: Find the derivative of (3x − 4)⁴ + x² − 7.

Question 10: Find the derivative of 3/(x+2)⁶.