Product Rule

The product rule is used to differentiate a function in Calculus. The product rule is employed when a function is the product of two or more functions. The derivatives of any two or more functions can be found using the Product Rule if the problems are a combination of two or more functions. D h(x) or h’ will be used to represent the derivative of a function h(x)

Product Rule

The product rule is a general concept that applies to problems involving differentiation, such as when one function is multiplied by another. The derivative of a product of two differentiable functions is equal to the first function multiplied by the derivative of the second, and the second function multiplied by the derivative of the first. The function could be exponential, logarithmic, or other.

Rules of Differentiation

Instead of using the generic approach of differentiation, differentiation rules allow us to evaluate the derivatives of specific functions. The attribute of linearity is important in the process of differentiation or finding the derivative of a function. For functions created from the fundamental elementary functions using the processes of addition and multiplication by a constant integer, this characteristic makes the derivative more natural.

Types of Differentiation Rules

The following are the most important differentiation rules:

• Power rule

• Product rule

• Sum and difference rule

• Chain rule

• Quotient rule

Product Rule Formula

The product rule formula is used to get the derivatives of two or more functions. Assume that u(x) and v(x) are two distinct functions. The product of the functions u(x)v(x) is differentiable in this way, and is written as (uv)′ = u′v + uv′.

The product rule formula provides an explanation to the conceptual theory that the product rule is given if the first function is multiplied by the derivative of the second function in addition to the second function multiplied by the derivative of the first function. In the first term, we use the constant ‘u,’ while in the second term, we use the constant ‘v.’

Use of Product Rule for Different Functions

The following is the product rule for various functions such as derivatives, exponents, and logarithmic functions:

Derivatives Product Rule

The product rule for any two functions, such as f(x) and g(x) is:

D[f(x) g(x)]

=f(x) D[g(x)] + g(x) D[f(x)]

d(uv)/dx =u(dv/dx) + v(du/dx)

u and v are two differentiable functions, respectively.

Product Rule for Exponents

If the numbers m and n are natural numbers, then xⁿx^m=x^n+m.

The product rule cannot be used to solve exponent expressions with distinct bases, such as 2^{3 ×}5⁴, or expressions likexnm. Only the Power Rule of Exponents may be used to solve an expression like xnm=xnm.

Product Rule for Logarithm

Let us take A and B that are positive real numbers with the base a.

Now, for this rule ‘a’ should not be equal to 0.

Therefore, the product rule would be: AB =A +B .

Triple Product Rule

The triple product rule is a broadening of the product rule. If three differentiable functions are f(x), g(x), and h(x), the product rule of differentiation can be applied to these three functions as follows:

Dfxgxhx=gxhxDfx+fxhxDgh+fxgxDhx

Conclusion

In calculus, the product rule is a method for determining the derivative of any function that is given in the form of a product produced by multiplying two differentiable functions. According to this rule the derivative of a product of two differentiable functions is equal to the sum of the product of the differentiation of the first function with the second function and the product of the differentiation of the second function with the first function.