Verifying differential equations (without substitution or integration) I am aware that several similar threads exist on this forum, however, my particular query is different from any previous question I've seen here. All  other answers have a 'y'term on both sides, and involve substitution as a means of verification. My question is a bit different. 
Basically, I want to know how I can verify this answer (without re-integrating it). 
Let's say a question was: 

$
\text{Find } \frac{dy}{dx} \text{ for } y = 4x^2 + 3x -6 
$

And my answer was: $\frac{dy}{dx} = 8x + 3$.
Is there any way to verify that my answer is correct (without using a calculator or otherwise integrating it)? I.e. any way to check that the derivative of $4x^2 + 3x -6$ is indeed $8x + 3$? If so, I would be most grateful if you could kindly explain the process of verification. 
Thanks in advance for your answers! 
Vignesh
[Edit] I did state that I needed a way to prove this without using a calculator, or any other similar electronic device. Essentially like in a non-calculator exam. Is there any way to prove this on paper?
 A: You can always resort to definitions and take the limit of the difference quotient
$$f'(x) = \lim_{h \to 0} \frac{4(x + h)^2 + 3(x + h) - 6 - (4x^2 + 3x - 6)}{h}$$
and check that it does indeed give $f'(x) = 8x + 3$, although you're far more likely to make an error here (and spend a nontrivial amount of time), compared to using the basic differentiation rules for polynomials carefully.
Without integration, I don't really know of any other methods to verify that your derivative is without a doubt correct. 
For something like this quadratic, you can get evidence by solving $f'(x) = 0$ and verifying that this happens at the $x$-coordinate of the vertex $\frac{-b}{2a}$, but this doesn't extend very well to other kinds of functions. It also doesn't guarantee a correct answer, it will just (potentially) show you if you're wrong.
A: If you just need to quickly check your work, you can plug it into 
Wolfram Alpha.
If you want, you can do this in either Mathematica

D[4*x^2+3*x+6,x]

or Python's SymPy package
from sympy import *
x = Symbol('x')
diff(4*x**2 + 3*x + 6, x)

