# Chain,product or quotient rule? why

• ibysaiyan
In summary, the chain rule can be used for functions that are composite, but it is not typically used for quotient functions.
ibysaiyan

## Homework Statement

Hi all well basically i have finished off chain rule and right now i am going through product rule and quotient, as i was going through some questions , i understood the basic rule and so on, but why i don't get is, how do i figure which rule i need to apply given equation using these three rules. For instance : y= x^3 sinx or y=x^3 / sinx , why would it be wrong to apply chain rule? Thanks for your replies ;) (grr i got to figure out using Latex =/)

## The Attempt at a Solution

For things in the form y=UV, use product rule

y=u/v use quotient rule (yes you can use the product rule as well, but using this directly is simpler.

You use the chain rule for when you have functions which have exponents that are difficult or tedious to expand, example:

y=(x+1)19712

clearly this is tedious, but the chain rule makes it easier to get the derivative.

ibysaiyan said:

## Homework Statement

Hi all well basically i have finished off chain rule and right now i am going through product rule and quotient, as i was going through some questions , i understood the basic rule and so on, but why i don't get is, how do i figure which rule i need to apply given equation using these three rules. For instance : y= x^3 sinx or y=x^3 / sinx , why would it be wrong to apply chain rule? Thanks for your replies ;) (grr i got to figure out using Latex =/)
The chain rule should be applied to composite functions, such as f(x) = sin(x^3). To evaluate f(b), for example, you first have to cube b, and then take the sine of that value.

y = x^3 * sin(x) is a product. Use the product rule.
y = x^3/sinx is a quotient. Use the quotient rule. Neither of these functions is composite, so it would be incorrect to apply the chain rule.

Sometimes you'll run across functions that seem likely candidates for a rule, but are not. For example, g(x) = x^2/5 is certainly a quotient. If you needed the derivative, you could use the quotient rule, but that's not advisable. Instead, think of this as (1/5)*x^2 and use the constant multiple rule, which says that d/dx(k*f(x)) = k*d/dx(f(x)). You should never use the quotient rule if the denominator is a constant. It's not that it will give you an incorrect derivative, but rather, that it's somewhat more complicated to use, and you are more likely to make a mistake. Even if you don't make a mistake, you are doing more work than you need to do, and life is short.

Oh,Thanks a lot you two.Now i get it.

## 1. What is the chain rule and why is it important in calculus?

The chain rule is a rule in calculus that allows us to find the derivative of a composite function. It states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function. This rule is important because many functions in calculus are composite functions and the chain rule allows us to easily find their derivatives.

## 2. How do you apply the chain rule to find the derivative of a function?

To apply the chain rule, you first need to identify the inner and outer functions of the composite function. Then, take the derivative of the outer function and multiply it by the derivative of the inner function. This will give you the derivative of the composite function.

## 3. What is the product rule and why is it useful in calculus?

The product rule is a rule in calculus that allows us to find the derivative of a product of two functions. It states that the derivative of a product of two functions is equal to the first function multiplied by the derivative of the second function plus the second function multiplied by the derivative of the first function. This rule is useful in calculus because it helps us find the derivatives of more complex functions.

## 4. How do you use the product rule to find the derivative of a function?

To use the product rule, you first need to identify the two functions being multiplied together. Then, take the derivative of the first function and multiply it by the second function. Next, take the derivative of the second function and multiply it by the first function. Finally, add these two terms together to get the derivative of the product function.

## 5. What is the quotient rule and when is it used in calculus?

The quotient rule is a rule in calculus that allows us to find the derivative of a quotient of two functions. It states that the derivative of a quotient of two functions is equal to the denominator multiplied by the derivative of the numerator minus the numerator multiplied by the derivative of the denominator, all divided by the square of the denominator. This rule is used when finding the derivatives of rational functions, where the numerator and denominator are both functions.

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