The product rule is a formal rule to find the derivatives of products of two or more functions.

In Leibniz’s notation we can express it as

\(\frac{d}{dx}(u . v) = \frac{du}{dx} . v + u . \frac{dv}{dx}\)OR

In Lagrange’s notation as,

\((u . v)’ = u’ v + u v’\)

This rule can be extended to a derivative of three or more functions.

Index

**History**

Gottfried Leibniz is credited with the discovery of this rule, he demonstrated it using differentials.

**Derivation**

**Leibniz’s Argument:**

Let \(u(x)\) and \(v(x)\) be two differentiable functions of x.

Then the differential of \(u.v\) is given by

\(d(u . v) = (u + du) . (v + dv) – u.v\)

\( = udv + vdu + du.dv\)

Since the term \(du.dv\) is negligible compared to \(udv + vdu\), as it becomes very small. So we neglect the term

Hence, Leibniz concluded that

\(d(u . v) = v . du + u . dv\)

dividing bot sides we come to

\(\frac{d}{dx}(u . v) = \frac{du}{dx} . v + u . \frac{dv}{dx}\)

**Proof**

Let us consider two differentiable functions \(f(x)\) & \(g(x)\), & \(h(x) = f(x)g(x)\).

Here, we will be proving that \(h\) is differential with \(x\) & \(h’(x)\) will be \(f’(x)g(x) + f(x)g’(x)\).

Factorising from first principle,

\(h(x) = f(x)g(x)\)

\(h’(x) = lim_{x \to 0} \frac{h(x + \Delta x) – h(x)}{\Delta x}\)

\(= lim_{x \to 0} \frac{f(x + \Delta x)g(x + \Delta x) – f(x)g(x)}{ \Delta x}\)

Adding and subtracting \(f(x)g(x + \Delta x)\) in numerator,

\(= lim_{x \to 0} \frac{ f(x + \Delta x)g(x + \Delta x) –f(x)g(x + \Delta x) + f(x)g(x + \Delta x) – f(x)g(x)}{ \Delta x}\)

\(= lim_{x \to 0} \frac{[ f(x + \Delta x) + f(x)] . g(x + \Delta x) + f(x) . [g(x + \Delta x) + g(x)]}{\Delta x}\)

\(= lim_{x \to 0} \frac{f(x + \Delta x) + f(x)}{\Delta x} . lim_{x \to 0} g(x + \Delta x) + lim_{x \to 0} f(x). lim_{x \to 0} \frac{g(x + \Delta x) + g(x)}{\Delta x}\)

Since, \(lim_{x \to 0} g(x + \Delta x) = g(x)\)

\(h’(x) = f’(x)g(x) + f(x)g’(x)\)

*Hence proved*.

**Application of Product Rule **

This rule is used mainly in calculus and is important when one has to differentiate product of two or more functions. It makes calculation clean and easier to solve.

**Examples**

**Question 1.** Differentiate the function: \((x^3 + 5)(x^2 + 1)\)

**Solution.** Here, \(f(x) = (x^3 + 5)\) & \(g(x) = (x^2 + 1)\)

Using this rule we get,

\(\frac{d (x^3 + 5) (x^2 + 1)}{dx} = \frac{d (x^3 + 5)}{dx}.(x^2 + 1) + (x^3 + 5). \frac{ d (x^2 + 1)}{dx}\)

= \((3x^2)(x^2 +1) + (x^3 + 5)(2x)\) = \(3x^4 + 3x^2 + 2x^4 + 10x\)

=> \(5x^4 + 3x^2 + 10x\).

**Question 2.** Find the derivative of \(h(x)\), if \(h(x) = f(x)g(x)\) & \(f(x) = sin(x)\) & \(g(x) = cos(x)\).

**Solution.** Given, \(h(x) = sin(x) . cos(x)\)

Therefore, \(h’(x) = cos(x)cos(x) + sin(x)(- sin(x))\)

\(\Rightarrow cos^2(x) – sin^2(x)\)

\(\Rightarrow cos(2x)\).

**FAQs**

**What is product rule?**It is a rule followed to differentiate the product of two functions,

\((xy)’ = x’y + xy’\)

**How we use product rule on three terms?**When there are three terms and this rule is to be applied, we group two functions and treat them as a single unit.

And hence apply the rule to left over two.

Ex. \((fgh)’ = (fg)’h + (fg)h’\)

\(= (f’g + fg’)h + fgh’\)

\(= f’gh + fg’h + fgh’\)

**What is difference between product rule and quotient rule?**Product rule is used to find the derivative of a product of function, while the quotient rule is used for finding derivative of function’s quotient.

**How is product rule different from Leibniz Rule?**Leibniz Rule is case of product rule in which we differentiate the product ‘n’ times.