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, 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 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 Derivatives

• 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}

Practice Questions

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

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

Question 3: Find the derivative of 2/x⁵.

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

Question 5: Find the derivative of (x+2)³ + 4x⁵ − x².

Question 6: Find the derivative of 8x⁷ − 5x³ + 2x.

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

Question 8: Find the derivative of 3/(x+2)⁶.

Question 9: Find the derivative of 6x⁴ + x² + 1/x³.

Question 10: Find the derivative of 9x⁵ + 2x³ − x + 4.