Product Rule Formula

Product rule is a fundamental principle in calculus used to differentiate the product of two functions. The rule states that the derivative of a product of two differentiable functions is given as, "product of first function with the derivative of second function added with differentiation of first function and second function."

In this article, we have covered product rule definition, product rule formula, product rule derivation, related examples and others in detail.

What is the Product Rule?

The product rule is a technique for determining the derivative of any function that is presented in the form of a product produced by multiplying two differentiable functions. The derivative of a product of two differentiable functions is equal to the sum of the product of the second function with differentiation of the first function and the product of the first function with differentiation of the second function, according to the product rule.

Product Rule Formula

If we have a function of type f(x)⋅g(x), we can use the product rule derivative to obtain the derivative of that function. The formula for the product rule is as follows:

d{u(x).v(x)}/dx = [v(x)×u′(x) + u(x)×v′(x)]

where,

• u(x) and v(x) are Differentiable Functions in R
• u'(x) and v'(x) are Derivatives of Functions u(x) and v(x) respectively

Derivation of Product Rule Formula

Suppose a function f(x) = u(x)⋅v(x) is differentiable at x. We will prove the product rule formula using the definition of derivatives or limits.

f'(x)=\lim_{\Delta x\to0} \frac{f(x+\Delta x)-f(x)}{\Delta x}

= \lim_{\Delta x\to0} \frac{u(x+\Delta x)⋅v(x+\Delta x)-u(x)⋅v(x)}{\Delta x}

= \lim_{\Delta x\to0} \frac{u(x+\Delta x)⋅v(x+\Delta x)-u(x)⋅v(x+\Delta x)+u(x)⋅v(x+\Delta x)-u(x)⋅v(x)}{\Delta x}

= \lim_{\Delta x\to0} \frac{(u(x+\Delta x)-u(x))⋅v(x+\Delta x)+u(x)⋅(v(x+\Delta x)-v(x))}{\Delta x}

= \lim_{\Delta x\to0} \frac{(u(x+\Delta x)-u(x))⋅v(x+\Delta x)}{\Delta x}+\lim_{\Delta x\to0}\frac{u(x)⋅(v(x+\Delta x)-v(x))}{\Delta x}

= v(x)\lim_{\Delta x\to0} \frac{u(x+\Delta x)-u(x)}{\Delta x}+u(x)\lim_{\Delta x\to0}\frac{v(x+\Delta x)-v(x)}{\Delta x}

Put \lim_{\Delta x\to0} \frac{u(x+\Delta x)-u(x)}{\Delta x}=u'(x)    and \lim_{\Delta x\to0}\frac{v(x+\Delta x)-v(x)}{\Delta x}=v'(x)

= v(x) × u'(x) + u(x) × v'(x)

This derives the formula for product rule.

Product Rule for Products of More Than Two Functions

Product rule is a fundamental principle in calculus used to differentiate the product of two functions. When dealing with more than two functions, the product rule can be extended to accommodate this complexity.

Product Rule for Three Functions

For three functions u(x), v(x)v, and w(x), the product rule is:

d{u(x).v(x).w(x)}​/dx = u'(x)×v(x)×w(x) + u(x)×v′(x)×w(x) + u(x)×v(x)×w'(x)

How to Apply Product Rule in Differentiation?

Product rule is a technique used in calculus to differentiate functions that are the product of two or more functions. Steps to apply the product rule is:

Step 1: Identify the functions u(x) and v(x) within the product.

Step 2: Differentiate each function separately.

Step 3: Apply the product rule formula.

Learn about the same in details with examples added below:

Read More:

• Quotient Rule
• Implicit Differentiation
• Advanced Differentiation
• Differentiation and Integration Formula

Practice Problems on Product Rule Formula

Problem 1: Find the derivative of the function f(x) = x sin x using product rule.

Solution:

We have, f(x) = x sin x.

Here, u(x) = x and v(x) = sin x

So, u'(x) = 1 and v'(x) = cos x

Using product rule we have,

f'(x) = v(x)u'(x) + u(x)v'(x)

= sin x (1) + x (cos x)

= sin x + x cos x

Problem 2: Find the derivative of the function f(x) = x log x using product rule.

Solution:

We have, f(x) = x log x.

Here, u(x) = x and v(x) = log x

So, u'(x) = 1 and v'(x) = 1/x

Using product rule we have,

f'(x) = v(x)u'(x) + u(x)v'(x)

= log x (1) + x (1/x)

= log x + 1

Problem 3: Find the derivative of the function f(x) = x² cos x using product rule.

Solution:

We have, f(x) = x² cos x

Here, u(x) = x² and v(x) = cos x

So, u'(x) = 2x and v'(x) = -sin x

Using product rule we have,

f'(x) = v(x)u'(x) + u(x)v'(x)

= cos x (2x) + x²(-sin x)

= 2x cos x - x² sin x

Problem 4: Find the derivative of the function f(x) = sin x log x using product rule.

Solution:

We have, f(x) = sin x log x

Here, u(x) = sin x and v(x) = log x

So, u'(x) = cos x and v'(x) = 1/x

Using product rule we have,

f'(x) = v(x)u'(x) + u(x)v'(x)

= log x (cos x) + sin x (1/x)

= log x cos x + sin x/ x

Problem 5: Find the derivative of the function f(x) = tan x sec x using product rule.

Solution:

We have, f(x) = tan x sec x

Here, u(x) = tan x and v(x) = sec x

So, u'(x) = sec² x and v'(x) = sec x tan x

Using product rule we have,

f'(x) = v(x)u'(x) + u(x)v'(x)

= sec x (sec² x) + tan x (sec x tan x)

= sec x (sec² x + tan² x)

= sec x (2sec² x - 1)

Problem 6: Find the derivative of the function f(x) = (x - 3) sin x using product rule.

Solution:

We have, f(x) = (x - 3) cos x

Here, u(x) = x - 3 and v(x) = sin x

So, u'(x) = 1 and v'(x) = cos x

Using product rule we have,

f'(x) = v(x)u'(x) + u(x)v'(x)

= sin x (1) + (x - 3) (cos x)

= sin x + x cos x - 3 cos x

Problem 7: Find the derivative of the function f(x) = x sec x using product rule.

Solution:

We have, f(x) = x sec x

Here, u(x) = x and v(x) = sec x

So, u'(x) = 1 and v'(x) = sec x tan x

Using product rule we have,

f'(x) = v(x)u'(x) + u(x)v'(x)

= sec x (1) + x (sec x tan x)

= sec x (1 + x tan x)

Problem 8. Find the derivative of the function f(x) = e^x cosx using the product rule.

Solution.

We have 𝑓(𝑥) = 𝑒^x cos𝑥

Here, 𝑢(𝑥)=𝑒^x and 𝑣(𝑥) = cos𝑥

So, 𝑢^, (𝑥) = (x)=e^x and 𝑣^,(x)= -sinx.

Using the product rule, we have:

f^,(x) = u^,(x)v(x) + u(x)v^, (x)

= e^x cosx + e^x (-sin x)

=e^x (cos x - sin x)

Problem 9. Find the derivative of the function f(x) = x³ sin x using the product rule.

Solution:

we have f(x) = x³ sinn x

here, u(x) = x³ and v(x) = sin x

so, u^,(x ) = 3x² and v^,(x) = cos x

using the product rule we have,

f^, (x) = u^,(x)v(x) + u(x)v^,(x)

= 3x² sin x + x³ cos x

Problem 10. FInd the derivative of the function f(x) = (3x² + 2x +1) tan x using the product rule.

Solution:

We have, f(x) = (3x² + 2x + 1) tan x

here, u(x) = 3x² + 2x +1 and v(x) tan x

so, u^,(x) = 6x +2 and v^,(x) = sec²x

using the product rule we have,

f^,(x) = u^,(x)v(x) + u(x)v^,(x)

= (6x +2) tan x + (3x² + 2x +1) sec²x

Related links:

• Product rule in derivative
• Application of derivative

Conclusion

The product rule is an essential method in calculus for finding the derivatives of products of multiple functions. Breaking down the product into manageable parts simplifies the process and makes finding the derivative easier. The equation

f^, (x) = u^,(x). v(x) + u(x).v^, (x)

shows how the derivatives of each function impact the total derivative of their multiplication. Comprehending and utilizing the product rule is crucial for solving intricate differentiation problems in different mathematical and real-life scenarios. Understanding this rule provides a greater understanding of functions and their rates of change, which is crucial in calculus.