# Rules of derivatives.

• Nov 10th 2013, 06:17 PM
sepoto
Rules of derivatives.
$\displaystyle \frac{-10}{(x+5)^3}$

Since this is a fraction I know I should use the quotient rule. My question is actually about the bottom part of the equation. I see it's to the third power and the quotient rule is going to be asking for g(x)' as its called and I start to look at the bottom and since it is to the third power wouldn't it be true that I would have to use the chain rule to find the derivative of (x+5)^3 which would be g'(x) for use in the quotient rule formula or am I going wrong?

• Nov 10th 2013, 06:20 PM
mikewezyk
Re: Rules of derivatives.
You don't *need* chain rule to find the derivative of g(x) since the x is to the first power and has no constant in front of it, you could just use power rule to get g'(x) to be 3(x+5)^2. You then HAVE to use quotient rule.
• Nov 10th 2013, 07:45 PM
sepoto
Re: Rules of derivatives.
I'm not sure I'm getting that yet. Is it really possible to just use the power rule. I was under the impression that (x+5)^3 would have to be expanded by the binomial theorem or alternatively the chain rule would have to be applied. The term I question has two terms parenthesized to the third power. I was under the impression that the power rule works on individual terms only.

I think what your saying is that since x is to the first power and has no constant in front of it that the power rule can still apply but otherwise it couldn't.
• Nov 10th 2013, 08:23 PM
topsquark
Re: Rules of derivatives.
Quote:

Originally Posted by sepoto
I'm not sure I'm getting that yet. Is it really possible to just use the power rule. I was under the impression that (x+5)^3 would have to be expanded by the binomial theorem or alternatively the chain rule would have to be applied. The term I question has two terms parenthesized to the third power. I was under the impression that the power rule works on individual terms only.

I think what your saying is that since x is to the first power and has no constant in front of it that the power rule can still apply but otherwise it couldn't.

Use the chain rule...it's easier this way. We have that g(x):
$\displaystyle g(x) = \frac{-10}{(x + 5)^3} = -10 \cdot (x + 5)^{-3}$

$\displaystyle g'(x) = (-10) \cdot -3 (x + 5)^{-4} \cdot (1)$

The 1 at the end is the derivative of x + 5.

-Dan
• Nov 11th 2013, 06:33 AM
HallsofIvy
Re: Rules of derivatives.
Mikewezyk's point is that, in using the chain rule, you don't have to write out every step. Taking u= x+ 5, $\displaystyle (x+ 5)^3= u^2$. By the chain rule, the derivative of $\displaystyle (x+ 5)^3$ is $\displaystyle 3u^2\frac{du}{dx}= 3(x+5)^2(1)$. Mikewezyk's point is that the derivative of x+ 5 is just 1. I would not agree that this is "not using the chain rule" but rather using the chain rule without writing everything out.

More generally, if g(x)= f(ax+ b) then taking u= ax+ b, g(x)= f(u) so that, by the chain rule, g'(x)= f'(u) du/dx= a f'(u). But with a little practice, you can do that "in your head". The derivative of the inear term ax+ b is a so that g'(x)= af'(ax+ b).