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Product Rule

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.

History

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

Gottfried Leibniz discovered Product Rule
Gottfried Leibniz (Source)

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.

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