In this page product rule for differentiation we are going to see one of the method of differentiating the function which are multiplying .

**(UV)' = UV' + VU'**

**Example 1:**

Differentiate x⁵ tan x

**Solution:**

Let y = x⁵ tan x

Here both function are multiplying now we have to consider one of the function as "u" and another as "v".

u = x⁵ v = tan x

u' = 5x⁴ v' = sec² x

Formula of product rule

**(UV)' = UV' + VU'**

= (x⁵)sec² x + (tan x)(5x⁴)

= x⁵sec² x + 5x⁴tan x

= x⁴[xsec² x + tan x]

**Example 2:**

Differentiate (x² + 7x +2) (x³ - log x)

**Solution:**

Here two functions are multiplying so we have to consider one of the function as "u" and another as "v"

u = (x² + 7x +2) v = (x³ - log x)

u' = 2x + 7(1) + 0 v' = 3x² - (1/x)

u' = 2x + 7 v' = 3x² - (1/x)

Formula of product rule

**(UV)' = UV' + VU'**

= (x² + 7x +2)[3x² - (1/x)] + (x³ - log x)(2x + 7)

**Example 3:**

Differentiate (x² - 1) (x² + 2)

**Solution:**

Here two functions are multiplying so we have to consider one of the function as "u" and another as "v"

u = (x² - 1) v = (x² + 2)

u' = 2x - 0 v' = 2x + 0

u' = 2x v' = 2x

Formula of product rule for differentiation

**(UV)' = UV' + VU'**

= (x² - 1)(2x) + (x² + 2)(2x)

= 2x³ - 2x + 2x³ + 4x

= 4x³ + 2x

This problem can be done by using another method.Here we have shown the alternate method without using product rule.

(x² - 1) (x² + 2)

= x²(x²) + 2x² - 1(x²) - 1(2)

= x⁴ + 2x² - x² - 2

= x⁴ + x² - 2

= d(x⁴) + d(x²) - d(2)

= 4x³ + 2x - 0

= 4x³ + 2x

**Related Topics **

**First Principles****Implicit Function****Parametric Function****Substitution Method****logarithmic function****Chain Rule****Quotient Rule****Rate of Change****Rolle's theorem****Lagrange's theorem****Finding increasing or decreasing interval****Increasing function****Decreasing function****Monotonic function****Maximum and minimum****Examples of maximum and minimum**

Quote on Mathematics

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life. They are:

It subtracts sadness and adds happiness in our life.

It divides sorrow and multiplies forgiveness and love.

Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”