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What is the Product Rule?

You know how to find the product of 2 numbers or expressions. But we don’t call it a product rule. We call it a product. We specifically use this product rule term in functions. Product rules help us in differentiating the product of two functions.

Let’s Learn Product Rule in Detail:

As we know the product rule tells us how to differentiate the product of two functions. Let F(x) be the product of 2 functions.

f(x) = p(x) × q(x) then, the derivative of f(x),

f′(x) = p′(x) × q(x) + p(x) × q′(x)

Where, f(x) = Product of differentiable functions p(x) and q(x), p(x) and q(x) = Differentiable functions, p'(x) = Derivative of function p(x), and q'(x) = Derivative of the function q(x).

This can also be written in Leibniz Notation as,

Product rule can be used to differentiate the product of three functions also.

Example: For three functions, r(x), s(x), and t(x), the product given as r(x)s(x)t(x), we have, (rst)’ =  r’st + rs’t  + rst’.

It can also be written in Leibniz Notation as


Let us briefly learn calculus and differentiation. Calculus is a branch of mathematics used to study continuous changes. It is applicable in many fields like maths, engineering, statistics, etc. For more information and worked examples on these topics, you may log on to the Cuemath website

There are two branches of calculus: differential calculus and integral calculus.

  • Differential Calculus slices something into little parts to observe how it changes.
  • Whereas Integral Calculus joins (integrates) the little pieces together to find how much there is.

Product Rule for Various Functions:

  1. Product Rule for Exponential functions: If p and q are the natural numbers, then xp × xq = xp+q. The Product rule cannot be used to solve expressions like x2 y3 where the bases are different and expressions like (xp)q. An expression like (xp)q can be solved only with the help of the Power Rule of Exponents where (xq)p = xqp.
  2. Product Rule for Logarithmic functions:

For any positive real numbers P and Q with the base a where a cannot be equal to zero,

logaPQ = logaP + logaQ

  1. Zero Product Rule: Zero product rule states, the product of two non-zero numbers results in zero only if one of the numbers is zero. If p and q are two numbers then pq = 0 only either p = 0 or q = 0.

if (x – a). x = 0, either x – a = 0 or x = 0

If x – a = 0, then x = a

So, the values of x are 0 and a. They are also called roots of the equation.

  1. Product Rule for Partial Derivatives: To find the partial derivatives of a function h such that h = a(x,y) b(x, y) we use the following formula.

Hx = and

Hy =

Example: Differentiate y =  sin  cos

Solution: Given: y = sin cos=

While differentiating, it becomes

= (sin ) [ ] + (cos ) [ ]

Differentiate the terms,  = sin  (- sin ) + cos  (cos )

= – sin2  + cos2

= (cos2  – sin2 )

But cos2  – sin2  = cos2  as per the identity

∴  = cos 2

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