Derivatives of Polynomial Functions (original) (raw)

Last Updated : 31 Oct, 2025

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

\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 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

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

**Some Basic Rules

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

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

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

**Solution:

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

This can be written as

_y = __x_−7

\frac{\partial y}{\partial x} = (-7) 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 + 6__x_3 − 4?

**Solution:

Let y = (_x + 5)2 + 6__x_3 − 4

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

**Example 4: Find the derivative of 6__x_3 + (6__x_ + 5)2 − 8__x_?

**Solution:

Let y = 6__x_3 + (6__x_ + 5)2 − 8__x_

\frac{\partial y}{\partial x} = 18x^{2}+2(6x+5)(6)-8\\ \frac{\partial y}{\partial x} =18x^{2}+12(6x+5)-8\\\frac{\partial y}{\partial x} =18x^{2}+72x+60-8\\\frac{\partial y}{\partial x} =18x^{2}+72x\ + 52

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

**Solution:

Let \ y=\frac{1}{(2x+8)^{7}}\\ y=(2x+8)^{-7}\\ \frac{dy}{dx}=-7(2x+8)^{-8}×\frac{d}{dx}(2x+8)\\ \frac{dy}{dx}=-7(2x+8)^{-8}×2\\ \frac{dy}{dx}=(-14)(2x+8)^{-8}

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.