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Product Rule in Derivatives

Last Updated : 31 Mar, 2024
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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.

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 product 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 proved 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) = limx→0{h(x + â–³x) – h(x)}/â–³x

= limx→0{a(x + â–³x).b(x + â–³x) – a(x).b(x)}/â–³x

= limx→0{a(x + â–³x).b(x + â–³x) – a(x).b(x + â–³x) – a(x).b(x)}/â–³x

= limx→0{[a(x + â–³x) – a(x)].b(x + â–³x) – a(x).[b(x + â–³x) – b(x)]}/â–³x

= limx→0{[a(x + â–³x) – a(x)].b(x + â–³x)}/â–³x – limx→0[a(x).[b(x + â–³x) – b(x)]}/â–³x

= {limx→0{a(x + â–³x) – a(x)}/â–³x}.{limx→0b(x + â–³x)} + {limx→0{b(x + â–³x) – b(x)}/â–³x}.{limx→0a(x + â–³x)}

= b(x).{limx→0{a(x + â–³x) – a(x)}/â–³x} + {limx→0{b(x + â–³x) – b(x)}/â–³x}.a(x)

Now,

  • {limx→0{a(x + â–³x) – a(x)}/â–³x} = a'(x)
  • {limx→0{b(x + â–³x) – b(x)}/â–³x} = 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)

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

Examples on Product Rule

Example 1: Find the derivative of the function y = exsinx

Solution:

y = ex.sinx

By Using Product Rule

y′(x) = (exsinx)′

= (ex)′sinx + ex(sinx)′ 

= exsinx + ex(cosx)

= ex(sinx + cosx)

Example 2: Find the derivative of the function Z= (y³ + 2y²-y)(eʸ – 1 ) .

Z=(y³ + 2y²-y)(eʸ – 1 )

Zʹ(x)=((y³ + 2y²-y)(eʸ – 1 ) ) ʹ

=(y³ + 2y²-y)(eʸ – 1 ) ʹ + (y³ + 2y²-y) ʹ (eʸ – 1 )

=(y³ + 2y²-y)(eʸ ) + ( 3y² + 4y -1 )(eʸ – 1 )

=(y³ + 5y²-3y-1 )(eʸ) -( 3y² + 4y -1 )

Example 3 : Find the derivative of the function y=x23x .

Given y=x23x

f(x)=x2 and f'(x) = 2x

g(x) =3x and g'(x) =3xlog3

Now

y’ =f(x)g'(x) + f'(x)g(x)

=x23xlog3 + 2×3x

=3xx(xlog3 + 2)

Example 4 : Find the differentiation of y=ex(cosx-sinx).

y=ex(cosx-sinx)

Let f(x) =ex then f'(x) = ex

and g(x) =cosx – sinx , then g’ (x) =-sinx -cosx

So

y’ = f(x) g'(x) + f'(x) g(x)

=ex(-sinx – cosx ) + ex (cosx – sinx)

=-2sinxex

Example 5 : Find the derivative of y =u.v.w , where u,v,w,y are function of x.

Given y = u.v.w

let f(x) = u and g(x) = v.w

then f'(x) = u’ and g'(x) =vw’ + v’w

So

y’ = f(x)g'(x) + f'(x) g(x)

=u(vw’ + v’w ) + u'(vw)

Example 6 : Find the derivative of y =sinx.cosx

Given y =sinx.cosx

Let f(x) =sinx and g(x) =cosx

then

f'(x) =cosx and g'(x) = -sinx

So

y’ = f(x)g'(x) + f'(x)g(x)

=sinx . (-sinx) + cosx . (cosx)

=cos2x – sin2x

Using Identity cos2x = cos2x – sin2 x

y’ = cos2x

FAQs on Product Rule

What is Product Rule of Differentiation in Calculus?

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

What is Product Rule Formula?

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

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

What is the Use of Product Rule in Differentiation?

The product rule in differentiation is used for various purposes,

  • It is used to find the differentiation 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.

What is Quotient 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}

How the product rule is related to chain Rule ?

The chain rule is used for differentiating composite functions, where one function is nested inside another. The product rule can be seen as a special case of the chain rule where the composite function is simply multiplication.

Are there any limitation of product rule ?

The product rule assumes that both functions being multiplied are differentiable. If one of the functions isn’t differentiable at a certain point, the product rule can’t be applied directly at that point.



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