# Differentiating a Polynomial in brackets.

I have the question below:

My question here is as you can see I have dropped the power of two down in-front of the bracket and reduced the power. Is this way of solving the question incorrect? I have a feeling what I should of done is expand the brackets and then differentiate the individual variables?

Why can I not use the chain rule on this?

• This is in no way a differential equation. – Captain Lama Mar 6 '16 at 18:33

Following the chain rule for $h(x)=f(x)^2$ we have $h'(x)=2f'(x)f(x)$. Hence this equals $2f(x)$ only if $f'(x)=1$, i.e., $f(x)$ is of the form $x+c$. However, here you have $f(x)=-x+c$. Long story short, if you do expand the expression, you should see that the correct result is indeed significantly (pun!) different.

• Hi @hagen-von-eitzen I still don't understand why I can't use the chain rule in this case. Is it because of the number? If so in what situations should I never use the chain rule and just expand the bracket? – Code Mar 6 '16 at 18:38
• You can use the chain rule, but you have to use it correctly. With $f(x)=20-x$ we have $f'(x)=-1$, so $h'(x)=2f'(x)f(x)=2\cdot(-1)\cdot (20-x)$. – Hagen von Eitzen Mar 6 '16 at 18:47
• Oh I see now. The mistake I made was that I did not differentiate the 20-X. I had just brought the power down and multiplied by that. So to clarify I can use the chain rule in any question that involves X in the bracket? (like this) – Code Mar 6 '16 at 18:52