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

[JessicaJ283782]

For example,

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

Thank you!

 

[Mark44]

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 6x^5/2, though, now it's no longer a composite, so you could use the power rule (and also the constant multiple rule).

 

[HallsofIvy]

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