Solving a Difficult Calculus Problem: Applying the Product Rule

Thread starter exmachina
Start date Dec 17, 2009
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Calculus Product Product rule

In summary, the Product Rule is a formula used in calculus to find the derivative of two functions that are multiplied together. It should be used when differentiating expressions involving products or variables raised to different powers. To apply the Product Rule, identify the two functions and use the formula "f'(x)g(x) + f(x)g'(x)" with their derivatives. Common mistakes to avoid include forgetting to take the derivative of one function and mixing up the order of the functions in the formula. The Product Rule can also be extended to more than two functions by adding additional terms to the formula.

Dec 17, 2009

exmachina

Homework Statement

http://i.imgur.com/TSLwA.png

Homework Equations

The Attempt at a Solution

Sorry I figured it out, it was the product rule.

Last edited: Dec 17, 2009

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Dec 17, 2009

Päällikkö

Homework Helper

It's not zero. That part is an identity from vector calculus, namely:
[tex]
\nabla \cdot (\psi\mathbf{A}) = \mathbf{A} \cdot\nabla\psi + \psi\nabla \cdot \mathbf{A}
[/tex]
as in http://en.wikipedia.org/wiki/Vector_calculus_identities

FAQ: Solving a Difficult Calculus Problem: Applying the Product Rule

What is the Product Rule in Calculus?

The Product Rule is a formula used in calculus to find the derivative of two functions that are multiplied together. It states that the derivative of a product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function.

When should I use the Product Rule?

The Product Rule should be used when you have a function that is a product of two other functions, and you need to find its derivative. It is also useful when differentiating expressions that involve variables raised to different powers, such as x^2 * y^3.

How do I apply the Product Rule in a calculus problem?

To apply the Product Rule, you must first identify the two functions that are being multiplied together. Then, use the formula "f'(x)g(x) + f(x)g'(x)" where f'(x) is the derivative of the first function and g'(x) is the derivative of the second function. Substitute the functions and their derivatives into the formula and simplify to find the derivative of the product.

What are common mistakes to avoid when using the Product Rule?

One common mistake is forgetting to take the derivative of one of the functions. Make sure to find the derivative of both functions and include them in the formula. Another mistake is mixing up the order of the functions in the formula, which will result in an incorrect answer. Be sure to follow the correct order of "f'(x)g(x) + f(x)g'(x)" when using the Product Rule.

Can the Product Rule be applied to more than two functions?

Yes, the Product Rule can be extended to more than two functions. For example, when differentiating f(x)g(x)h(x), you would use the formula "f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x)". Each term in the formula represents the derivative of one of the functions multiplied by the remaining functions. This can be extended to any number of functions being multiplied together.