Product Rule Calculator

The product rule calculator helps you to calculate the derivation of product rule and find the differentiation of product rule of two or more than two functions easily.

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Introduction to Product Rule Calculator:

Product rule calculator with steps is the best online source that is used to evaluate the derivation of product rule problems. It helps you to find the differentiation of the product rule of two or more than two function.

Product Rule Calculator with Steps

Product rule derivative calculator is a useful tool for students, teacher, professional or researcher who wants to find the product rule derivation function solution without doing manual calculation.

What is the Product Rule?

Product rule is define as the process in calculus that is used to find the derivation of product of two or more than two functions. For example, the first function is f(x) and the second function is g(x) the product rule of f(x).g(x) for differentaition given by the derivative calculator product rule is,

$$ (f(x) . g(x))’ \;=\; f’(x) . g(x) + f(x) . g’(x) $$

f(x).g(x) is the product rule function and its differentiation is (f(x).g(x))`. This rule is essential in mathematic because it used in various field other than mathematics like economic,physic or engineering etc.

How to Find Derivative Product Rule?

The product rule is a fundamental concept in calculus used to differentiate product of functions. It provides a systematic method to find the derivative function that does not solve with direct differential rule. Here’s a step-by-step guide on how to apply the product rule:

Step 1:

Identify the two functions that is being multiplied. Suppose you have a product of functions f(x) and g(x).

$$ h(x) \;=\; f(x) ⋅ g(x) $$

Step 2:

Differentiate each function f(x) and g(x) with respect to x separately. Find f′(x), the derivative of f(x) and g′(x), the derivative of g(x).

Step 3:

Use the product rule formula and add the value of f`(x) and g`(x),

$$ (f(x) . g(x))’ \;=\; f’(x) . g(x) + f(x) . g’(x) $$

Step 4:

Simplify the expression after adding the value of f(x) and g(x) and its derivation value respectively

Step 5:

After simplification you get the solution of your given product rule derivation function.

By following these steps, you can easily apply the product rule to find the derivative of the product of functions.

Practical Example of Product Rule:

The example of product rule derivation problem will help you in the conceptual clarity of calculation process of product rule calculator with steps.

Example: Differentiate the following:

$$ y \;=\; x^{-2} (4 + 3x^{-3}) $$

Solution:

Differentiate with respect to x,

$$ y’ \;=\; x^{-2} D[4 + 3x^{-3}] + D[x^{-2}] (4 + 3x^{-3}) $$

$$ =\; x^{-2} (-9x^{-4}) + (-2x^{-3})(4 + 3x^{-3}) $$

$$ =\; -9x^{-6} - 8x^{-3} - 6x^{-6} $$

$$ =\; -15x^{-6} - 8x^{-3} $$

$$ =\; \frac{-15}{x^6} - \frac{8}{x^3} $$

$$ =\; \frac{-15}{x^6} - \frac{8x^3}{x^6} $$

$$ =\; -\frac{15 + 8x^3}{x^6} $$

How to Use the Product Rule Calculator?

The derivative calculator product rule has an easy-to-use interface, so you can easily use it to evaluate the derivative of a given product function. Before adding the input for the solutions of product, you must follow some simple steps. These steps are:

Add the product rule problems that you want to differentiate in the input box.
Choose the variable of derivation for product rule in its respective field.
Recheck your input value for product rule derivation before hitting the calculate button to start the calculation process in the product rule derivative calculator.
The “Calculate” button gives the desired result of your given product rule problem.
If you want to try out our product rule solver to check the accuracy in solution then use the load example for clarity.
The “Recalculate” button brings you to a new page for solving more product rule derivation questions.

Outcome from Product Rule Derivative Calculator:

The derivative calculator product rule with steps gives you the solution to a given differential problem when you give the input and it provides you with solutions. It may contain as:

Result Option:

You can click on the result option and it provides you with a solution of product rule derivation questions

Possible Step:

When you click on the possible steps option then the product rule calculator with steps provides you with the solution of product rule of differential problem in steps.

Benefits of Derivative Calculator Product Rule:

The product rule calculus calculator gives you multiple benefits whenever you use it to calculate product rule of given function and you get its solution. These benefits are:

The product rule derivative calculator saves the time and effort that you consume in solving complex product rule questions.
It is a free-of-cost tool that provides you with a solution for a given product rule problem to find its result without spending anything.
The product rule solver is an adaptive tool that allows you to evaluate different types of product rule functions (logarithmic, exponential, algebraic, inverse trigonometry, etc).
You can use this calculator for practice to get a strong hold on this concept.
It gives you accurate solution of product rule differential problems without any difficulty.
Product rule Calculator with steps can operate on an online platform which means you can use it anytime.

Related References

Frequently Ask Questions

What is the difference between chain rule and product rule

The chain rule and the product rule are both important rules in calculus for finding the derivatives of functions, but they are used in different situations. Here's a detail about how they differ:

Chain Rule:

The chain rule is used to differentiate composite functions. It helps you find the derivative of a function that is composed of two or more functions.

Mathematical Statement:

$$ \frac{d}{dx} [f(g(x))] \;=\; f’ (g(x)) . g’(x) $$

Product Rule:

The product rule is used to differentiate the product of two functions. It helps you find the derivative of a function that is the product of two other functions. Mathematical Statement is:

$$ \frac{d}{dx} [u(x) . v(x)] \;=\; u’(x) . v(x) + u(x) . v’(x) $$

You differentiate the first function and multiply by the second function, then add the product of the first function and the derivative of the second function.

When to use product rule vs chain rule

The product rule and chain rule are both used for finding the derivatives of functions, but they are used in different ways. Here’s a detail on when and how to use each rule.

Use the product rule when you need to find the derivative of a function that is the product of two or more functions.

If you have h(x)=u(x).v(x)

$$ h`(x) \;=\; u`(x) . v(x) + v`(x) . u(x) $$

If you have a function where two separate functions are multiplied, then you can apply the product rule.

Use the chain rule when you need to find the derivative of a composite function that one function is inside another function.

If h(x) = f(g(x)) h(x) then,

$$ h′(x) \;=\; f′(g(x)) ⋅ g′(x) $$

When you have a function composed of another function, then you can apply the chain rule.

Does derivative product rule work for negative exponents

Yes, the product rule for derivatives works for functions with negative exponents, as it does for functions with positive exponents. The presence of negative exponents does not change the fundamental rules of the product rule. However, it may require some algebraic expression and simplification for these types of given expressions.

Why not use the product rule for derivative of 5sinx

The product rule is not used for 5sin⁡(x) because this function is a constant times a single function, and the constant multiplication rule is more straightforward than any other method. The product rule is applied only when you have the product of two different functions.

How to calculate a triple product rule

To the calculation of the triple product rule, let's take an example of the triple product rule with a solution to know its working process.

Example:

Find the derivative of f(x) = (x^2 + 1) . sin(x) . e^x

Solution:

Apply triple product rule formula,

$$ \frac{d}{dx} [u(x) . v(x) . ω(x)] \;=\; u’(x) . v(x) . ω(x) + u(x) . v’(x) . ω(x) + u(x) . v(x) . ω’(x) $$

Differentiation with respect to x,

$$ u(x) \;=\; x^2 + 1,\; u’(x) \;=\; 2x $$

$$ v(x) \;=\; sin(x), v’(x) \;=\; cos(x) $$

$$ ω(x) \;=\; e^x, ω’(x) \;=\; e^x $$

$$ \frac{d}{dx} [(x^2 + 1) . sin(x) . e^x] \;=\; (2x . sin(x) + (x^2 + 1) . cos(x) + (x^2 + 1) . sin(x)) . e^x \;=\; (2x sin(x) + cos(x)(x^2 + 1) + sin(x)(x^2 + 1)) . e^x $$

Amanda Jepson

Last Updated February 21, 2024