# 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.**

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**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 x ^{3}?**

**Solution:**

Let y = x

^{3}

**Some basic rules of derivative**

- If y = c f(x)

- If y = c

**Example 1: Find the derivative of 4x ^{3 }+ 7x?**

**Solution:**

Let y = 4x

^{3 }+ 7x

**Example 2: Find the derivative of 3x ^{2 }– 7?**

**Solution:**

Let y = 3x

^{2 }– 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 7x ^{5 }+ x^{3} − x?**

**Solution:**

Let y = 7x

^{5 }+ x^{3}− x

**Example 3: Find the derivative of ****(****x ****+ 5) ^{2 }+ 6**

**x**

^{3 }− 4?**Solution:**

Let y = (

x+ 5)^{2 }+ 6x^{3 }− 4

**Example 4: Find the derivative of ****6****x**^{3} + (6**x ****+ 5) ^{2} − 8**

**x****?**

**Solution:**

Let y = 6

x^{3}+ (6x+ 5)^{2}− 8x

**Example 5: Find the derivative of ****?**

**Solution:**