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.

Polynomials are always continuous and differentiable $\forall x \in R$
Here use $\frac{d x^n}{dx}=n x^{n-1}$, Also use the distributivity of $\frac{d}{dx}$ as :$\frac{d}{dx}[ a f(x)+ b g(x)] =a\frac{d}{dx} f(x)+ b\frac{d}{dx} g(x)$, to get
$$\frac{d}{dx}(4x^7-14x^4+30 x-17)= 4 \frac{d}{dx} x^7-14 \frac{d}{dx} x^4 +30\frac{d}{dx}x- \frac{d}{dx}17= 28 x^6-56 x^3 +30. $$
