Product Rule Formula: Derivation & Differentiation

Product rule formula enables us to distinguish between two or more functions within a given function. The formula for the product rule is as follows: 

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

• The product rule is revealed when multiplying the first function by the derivatives of the second plus the second function multiplied by the derivatives of the first function. 
• Here, we assume that the first term of u constant and the second term of v constant are constants.
• The Product Rule, or the Leibniz Rule, can be applied to determine the derivative of a given function of the form, f(x).g(x), in order for both f(x) and g(x) to become differentiable.
• It follows the very concept seen in limits and derivatives.

Key Terms: Differentiation, Derivative, Limits, Product Rule, Leibniz Rule, Chain Rule, Lagrange’s Notation

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What is Product Rule?

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"Product rule" in mathematics is used to determine the differential of any function presented in the form of a product formed by multiplying any two differentiable functions. 

• The derivation of a product of two differentiable functions is equal to the sum of the first product function with the second function differentiation and the first product function with the second function differentiation, as per product rule.
• This indicates that if a function of the format f(x).g(x) is given, we can use the product rule derivative to calculate the derivation of this function as, d / dx f(x)·g(x) = [g(x) × f'(x) + f(x) × g'(x)]

Discover about the Chapter video:

Continuity and Differentiability Detailed Video Explanation:

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Product Rule Formula

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The Product Rule Formula can be expressed as:

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

Using the mathematics product rule formula, we can calculate the derivatives or assess the differentiation of the product of two functions. The following is the product rule formula of two functions:

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

Here,

• f(x) stands for Product of differentiable functions u(x) and v(x)
• u(x), v(x) stands for Differentiable functions
• u'(x) stands for Derivative of function u(x)
• v'(x) stands for Derivative of the function v(x)

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Derivation of Product Rule Formula

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The product rule can be expressed using Lagrange notation for any two functions as

In Leibniz's notation, this is written as (u v)' = u'·v + u·v'.

d / dx (u·v) = du / dx ·v + u·dv / dx

Let's examine this evidence for the product rule formula. There are various ways to demonstrate the product rule formula, including:

• Using the First principle

• With the Chain rule

Product Rule Formula using First Principle

Let the function h(x) = f(x)g(x), such that f(x) and g(x) are differentiable at x, be used to demonstrate the product rule formula using the definition of derivative or limits.

 h'(x) = lim_Δx→0 [h(x + Δx) - h(x)]/Δx

= lim_Δx→0 f(x+Δx)g(x+Δx)−f(x)g(x) / Δx

= lim_Δx→0 f(x+Δx)g(x+Δx)−f(x)g(x+Δx)+f(x)g(x+Δx)−f(x)g(x) / Δx

= lim_Δx→0 [f(x+Δx)−f(x)]g(x+Δx)+f(x)[g(x+Δx)−g(x)] / Δx

= lim_Δx→0 [f(x+Δx)−f(x)]g(x+Δx) / Δx + lim_Δx→0 f(x)[g(x+Δx)−g(x)] / Δx 

= (lim_Δx→0 [f(x+Δx)−f(x)] / Δx) (lim_Δx→0 g(x+Δx))+(lim_Δx→0 f(x)) (lim_Δx→C

= g(x)lim_Δx→0 [f(x+Δx)−f(x)] / Δx + f(x) lim _Δx→0 [g(x+Δx)−g(x)] / Δx 

∵ lim_Δx→0 [f(x+Δx)−f(x)] / Δx lim_Δx→0 [g(x+Δx)−g(x)] / Δx = g'(x)

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

Hence, proved.

Product Rule Formula using Chain Rule

By thinking of the product rule as a specific case of the chain rule, you can use the chain rule formula to obtain the mathematics formula for the product rule. If f(x) is a differentiable function, then h(x) must equal f(x)g (x).

d / dx (f·g) = [δ(fg) / δf][df / dx] + [δ(fg) / δg][dg / dx] = g(df / dx) + f(dg / dx)

Product rule for Product of more than Two Functions

Through the same technique, the product rule can be expanded to encompass products of more than two components. For instance, assuming that the product of three functions, u(x), v(x), and w(x), is considering as u(x)v(x)w(x), we obtain,

d(uv w) / dx = vw du/dx + u dv/dx w + uv dw/dx d(uvw) / dx = du/dx vw + u d/d

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Product Rule in Differentiation

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In order to determine the derivative of the function h(x) = f(x)g(x), both f(x) and g(x) have to be differentiable functions. By using the product rule, we can apply the following steps to find the derivation of a differentiable function h(x) = f(x)g(x):

Step 1: Check the values of f(x) and g(x).

Step 2: Now, determine the values of f'(x) and g'(x) and further apply the product rule formula, which can be expressed as: h'(x) = \(\frac{d}{dx}\)f(x)·g(x) = [g(x) × f'(x) + f(x) × g'(x)]

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Things to Remember

1. A mathematics technique known as the "product rule" can be used to determine the derivative or differentiation of a function that is given as the combination of two differentiable functions.
2. The product rule effectively adopts the differentiations notions of limits and derivatives.
3. The product rule of differentiation or (dy/dx) = u (dv/dx) + v (du/dx) is also known as the product rule for derivatives.
4. Gottfried Leibniz's product rule enables us to compute derivatives that we don't want (or are unable to) multiply swiftly.

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