Product Rule in Derivatives

Product Rule is the rule that is used to find the derivative of the function that is expressed as the product of two functions. The product rule in calculus is the fundamental rule and is used to find the derivative of the functions. Suppose we have a function that is a product of the two functions such that, f(x) = a(x).b(x)then the derivative of the function is represented as

f'(x) = a(x).b'(x) + a'(x).b(x)

where,

• f'(x) is the derivative of f(x)
• a'(x) is the derivative of a(x)
• b'(x) is the derivative of b(x)

Product Rule of the calculus is proved using the concept of limit and derivatives. In this article, we will learn about the Product rule, the product rule formula, its proof, examples, and others in detail in this article.

What is Product Rule?

When the derivative of two or more functions is to be taken, the product rule is applied. The product rule states that if a function is the product of the two functions then the derivative of the function is the sum of the product of the first function and the derivative of the second function, with the product of the second function and the derivative of the first function. For any given function that is the product of the two functions,

d/dx{f(x)·g(x)} = [g(x) × f'(x) + f(x) × g'(x)]

Product Rule Formula

The product rule formula in calculus is the formula that gives the way to find the differentiation of two functions and the formula for the product rule formula is given as,

Suppose we have f(x) = u(x).v(x) then the differentiation of f(x) is find as,

d/dx{u(x)·v(x)} = [v(x) × u'(x) + u(x) × v'(x)]

where,

• u(x) and v(x) are the differential functions
• u'(x) is the derivative of u(x)
• v'(x) is the derivative of v(x)

Derivation of Product Rule Formula

Let us take two functions a(x) and b(x). So, the Product rule arrives when you multiply the first function a(x) with the derivative of the second function b(x) plus the derivative of the first function a(x) multiplied by the second function b(x). Thus we make the product rule as,

(ab)’ = a’b + ab’

In Leibniz’s notation, it is written as,

d/dx(u.v) = du/dx.(v) + (u).dv/dx

This formula can be used by two methods,

• Using First Principle
• Using Chain Rule

Now let’s prove the same by both methods,

Product Rule Formula Using First Principle Proof

Using the first principle of the derivative we can easily prove the product rule as, suppose we have a function h(x) = a(x).b(x) then its differentiation is found using,

h'(x) = lim_x–>0{h(x + △x) – h(x)}/△x

= lim_x–>0{a(x + △x).b(x + △x) – a(x).b(x)}/△x

= lim_x–>0{a(x + △x).b(x + △x) – a(x).b(x + △x) – a(x).b(x)}/△x

= lim_x–>0{[a(x + △x) – a(x)].b(x + △x) – a(x).[b(x + △x) – b(x)]}/△x

= lim_x–>0{[a(x + △x) – a(x)].b(x + △x)}/△x – lim_x–>0[a(x).[b(x + △x) – b(x)]}/△x

= {lim_x–>0{a(x + △x) – a(x)}/△x}.{lim_x–>0b(x + △x)} + {lim_x–>0{b(x + △x) – b(x)}/△x}.{lim_x–>0a(x + △x)}

= b(x).{lim_x–>0{a(x + △x) – a(x)}/△x} + {lim_x–>0{b(x + △x) – b(x)}/△x}.a(x)

Now,

• {lim_x–>0{a(x + △x) – a(x)}/△x} = a'(x)
• {lim_x–>0{b(x + △x) – b(x)}/△x} = b'(x)

h'(x) = b(x).a'(x) + a(x).b'(x)

Thus, the product rule is proved

Product Rule Formula Using Chain Rule Proof

Using the Chain Rule of the derivative we can easily prove the product rule as, suppose we have a function h(x) = a(x).b(x) then its differentiation is found using,

d/dx.{h(x)} = d/dx.{a(x).b(x)} = d/dx.{a.b}

= {d(a.b)/da}.{da/dx} + {d(a.b)/db}.{db/dx}

= b.{da/dx} + a{db/dx}

= a’.b + a.b’

Thus, the product rule is proved.

Product Rule for Products of More Than Two Functions

Product rule for more than two functions is simply found using the product of two functions. And then applying the product rule again,

d(a.b.c)/dx = da/dx.(b.c) + a.(db/dx).c + a.b.(dc/dx)(d{a.b.c}/dx)

= da./dx(b.c) + a.(db/dx).c + a.b.(dc/dx)

Example: Find the derivative of (x² + x)(3x + 5)

Solution:

We know that, f′(x) = X(x).Y′(x) + Y(x).X′(x)

X(x) = (x² + x), Y(x) = (3x + 5)

= (x² + x)'(3x + 5) + (x² + x)(3x + 5)’

= (2x + 1)(3x + 5) + (x² + x)(3)

= 6x² + 10x + 3x + 5 + 3x² + 3x

= 9x²+ 16x + 5

Applying Product Rule in Differentiation

Product rule is applied to the product of the function, follow the steps discuss below,

Step 1: Identify the function f(x) and g(x)

Step 2: Find the derivative functions f'(x) and g'(x)

Step 3: Use the formula,

d/dx{f(x).g(x)} = f(x).g'(x) + f'(x).g(x)

Then use the formula to get the required differentiation.

Read More,

• Differentiation and Integration Formula
• Advanced Differentiation
• Implicit Differentiation

Examples on Product Rule

Example 1: Differentiate y = cos²x using the product rule.

Solution:

y(x) = cosx.cosx

By Using Product Rule,

y′(x)= (cosx.cosx)′ = (cosx)′.cosx + cosx.(cosx)′

(cosx)′ = -sinx

y′(x) = -sinx.cosx + cosx(-sinx)

= -2sinxcosx

= -sin2x

Example 2: Find the derivative of the function y = e^xsinx

Solution:

y = e^x.sinx

By Using Product Rule

y′(x) = (e^xsinx)′

= (e^x)′sinx + e^x(sinx)′ 

= e^xsinx + e^x(cosx)

= e^x(sinx + cosx)

Examples 3: Find the derivative of the function y = x.sinx

Solution:

By Using Product Rule

y′(x) = (x.sinx)′

= (x)′sinx + x (sinx)′ 

= sinx + x cosx

Example 4: Find the derivative of the function y = x(1 + x)

Solution:

By Using Product Rule

y'(x) = {x(1 + x)}’

= x'(1 + x) + x(1 + x)’ 

= (1 + x) + x(0 + 1) 

= 1 + 2x

FAQs on Product Rule

1. What is Product Rule of Differentiation in Calculus?

The product rule of the differentiation is the rule used in calculus that is used to find the differentiation of the product of the two functions.

2. What is Product Rule Formula?

The product rule formula is the formula that is used to find the differntiation of two function, suppose we have to find the differentitaion of f(x) = h(x).g(x) such that,

f'(x) = h(x).g'(x) + g(x).h'(x)

3. What is the Use Product Rule in Differentiation?

The product rule in differentiation is used for various purposes,

• It is used to find the differntiation of the function that are expressed as the product of two functions.
• It is used to find the rate, maxima, minima, etc and others in detail, etc.

4. What is Quotuient Rule?

The quotient rule in the differentiation is used to find the differentiation of the function that are expressed as the division of two functions,

Suppose we have a function f(x) = g(x)/h(x) then the differentiation of f(x) is found as,

f'(x) = {h(x).g'(x) – g(x).h'(x)}/ {h(x)²}

5. What is the Differentiation of u.v?

The differentiation of u.v is found using the formula,

d/dx(u.v) = u.v’ + u’.v