In simple words, product means multiplying two functions together. We can use this rule to calculate derivatives we don’t want or can’t multiply quickly. In other words, the product rule combines our understanding of the power rule and the sum and difference rule for derivatives to allow us to compute the derivative of two differentiable functions that are multiplied together. The derivative of a product of two differentiable functions is equal to the sum of the second function’s product with differentiation of the first function and the first function’s product with differentiation of the second function.

## What do you Mean by Product Rule?

The product rule is a general concept that applies to problems involving differentiation, such as when one function is multiplied by another. The derivative of a product of two differentiable functions is equal to the first function multiplied by the derivative of the second, and the second function multiplied by the derivative of the first. The function could be exponential, logarithmic, or other.

So, an example would be y = x^{2} cos 4x So here we have one function, x^{2} , multiplied by a second function, cos 4x. Notice that we can write this as y = uv where u= x ^{2} and v = cos 4x.

## Product Rule Formula

**Product Rule for two functions**

According to the product rule differentiation, if the function f(x) is the product of any two functions, let’s say p(x) and q(x) here, then the derivative of the function f(x) is,If function f(x) =p(x) ×q(x) then, the derivative of

** f(x),f′(x) =p′(x) × q(x) + p(x) × q′(x)**

**Product Rule for three functions**

If the function f(x) is the product of three functions, let’s say p(x) ,q(x) and r(x) here, then the derivative of the function f(x) is, If function f(x) =p(x) × q(x)× r(x) then, the derivative of

**(pqr)’ = p’(x)q(x)r(x) + p(x)q’(x)r (x) + p(x)q(x)r’(x)**

**Product Rule for Exponent:**If m and n are the natural numbers, then

**x**Product rule cannot be used to solve expressions of exponent having a different base like a^{a}× x^{b}= x^{a+b }^{x}× b^{y}and expressions like (x^{a})^{b}. An expression like (x^{a})^{b}can be solved only with the help of the Power Rule of Exponents where (x^{a})^{b}= x^{ab}.**Product Rule for Logarithm**If P and Q are positive real numbers with the base is x

where, x≠ 0,**log**_{X}PQ = log_{X}P + log_{X}Q

## Examples

**Example 1:** Simplify the expression: y= x^{5} × x^{6}

**Solution:
**Given: y= x

^{5}× x

^{6 }product rule for the exponent is

**x**If you use the product rule, you can write it as follows:

^{a}× x^{b}= x^{a+b }y = x

^{5}× x

^{6 }y = x

^{11 }Hence, the simplified form of the expression, y= x

^{5}× x

^{6}=x

^{11}

**Example 2:** Differentiate the following functions: f(x) = (7x^{3} -x)(12 – 10x)

**Solution:**

We know the product rule for two functions:** f(x),f′(x) =p′(x) × q(x) + p(x) × q′(x)
**f′(x)=(21x

^{2}-1)(12-10x) + (7x

^{3}– x)(- 10)

Further simplifying we get

f′(x)=252x

^{3}– 280x

^{2}+ 20x-12

## Calculus

Calculus is a branch of mathematics that deals with change rates. Prior to the invention of calculus, all arithmetic was static, and it could only be used to calculate unmoving objects. In reality, everything in the universe is always moving. Calculus allowed scientists to calculate how particles, stars, and matter move and change in real-time. Physics, engineering, economics, statistics, and medicine are just a few of the domains where calculus is applied. Calculus is also employed in a variety of fields, including space flight, calculating how medication interacts with the body, and even designing safer architecture.

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