# Question about determining whether to use the chain rule or not?

• JessicaJ283782
In summary, for differentiating 6*sqrt(x^5), the chain rule is necessary as it is a composite function. However, it can also be simplified to 6x^{5/2} where the power rule can be used. It ultimately depends on the individual problem and using your knowledge to determine the most efficient method.
JessicaJ283782
For example,

if you differentiate 6*sqrt(x^5), would you use the chain rule? If not, why?

Thank you!

JessicaJ283782 said:
For example,

if you differentiate 6*sqrt(x^5), would you use the chain rule? If not, why?

Thank you!
Should you use chain rule? It depends.

As your function is written, you have a composite function (a function whose argument is another function). To differentiate such a function requires the chain rule.

If you write your function as 6x5/2, though, now it's no longer a composite, so you could use the power rule (and also the constant multiple rule).

It's not a matter of applying some "hard and fast" rule. You use your knowledge and think about each individual problem. Any time you can see something that can be thought of as a composition of two (or more) functions, that is candidate for the chain rule. To differentiate $6\sqrt{x^5}$ you can think of its as f(g(x)) where $f(x)= 6\sqrt{x}$ and $g(x)= x^5$.

In that case, $g'(x)= 5x^4$ and $f'(x)= (6x^{1/2})'= 6(1/2)x^{-1/2}= 3/\sqrt{x}$ so the derivative is $(3/\sqrt{x^5})(5x^4)= 15(x^{-5/2})(x^4)$$= 15x^{-5/2+ 4}= 15x^{3/2}$.

But, in this particular case, it is easier to do as Mark44 suggested: write the function as $6(x^5)^{1/2}= 6x^{5/2}$ so its derivative is $6(5/2)x^{5/2- 1}= 15x^{3/2}$ as above

## 1. What is the chain rule in calculus?

The chain rule is a calculus rule used to find the derivative of a composite function. It states that the derivative of the outer function multiplied by the derivative of the inner function.

## 2. How do I determine if the chain rule should be used?

The chain rule should be used when the given function is a composition of two or more functions, where the inner function is a function of the independent variable.

## 3. Can the chain rule be used for any type of function?

Yes, the chain rule can be used for any type of function, as long as it meets the criteria of being a composite function.

## 4. Is it necessary to use the chain rule every time a function is composed?

No, the chain rule is only necessary when taking the derivative of a composite function. If the function is not composed, the chain rule is not needed.

## 5. Can the chain rule be used for higher order derivatives?

Yes, the chain rule can be used for higher order derivatives. Each time the derivative is taken, the chain rule will need to be applied again.

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