<s>[INST] Differentiable function $f(x)=4x^7 −14x^4 +30x−17$ I am trying to prove that the function $f:\Bbb R→\Bbb R$, $f(x)=4x^7 −14x^4 +30x−17$,is injective. To do this I need to prove it is differentiable from first principles. I can then prove its derivative is strictly increasing to show it is injective. Any help on the proof that it's differentiable would be great, especially in what delta to choose.
 [/INST] The OP is asking for a first-principles derivation of the derivative  of a specific polynomial. Here's one way to do it, using the algebraic identity $(x^n-y^n)=(x-y)(x^{n-1}+x^{n-2}y^2+\cdots+xy^{n-2}+y^{n-1})$.
If $f(x)=4x^7-14x^4+30x-17$, then
$$\begin{align}
{f(x)-f(a)\over x-a}
&={4(x^7-a^7)-14(x^4-a^4)+30(x-a)\over x-a}\\
&={4(x-a)(x^6+x^5a+x^4a^2+x^3a^3+x^2a^4+xa^5+a^6)-14(x-a)(x^3+x^2a+xa^2+a^3)+30(x-a)\over x-a}\\
&=4(x^6+x^5a+x^4a^2+x^3a^3+x^2a^4+xa^5+a^6)-14(x^3+x^2a+xa^2+a^3)+30
\end{align}$$
hence
$$\begin{align}
f'(a)&=\lim_{x\to a}{f(x)-f(a)\over x-a}\\
&=\lim_{x\to a}(4(x^6+x^5a+x^4a^2+x^3a^3+x^2a^4+xa^5+a^6)-14(x^3+x^2a+xa^2+a^3)+30)\\
&=4(a^6+a^5a+a^4a^2+a^3a^3+a^2a^4+aa^5+a^6)-14(a^3+a^2a+aa^2+a^3)+30\\
&=4\cdot7a^6-14\cdot4a^3+30\\
&=28a^6-56a^3+30
\end{align}$$
</s>