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

(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

Some basic rules of derivative

• 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: