Find the derivative of x^2+6x+10 using the first principle .

First, we'll express the first principle of finding the
derivative of a given function:

lim [f(x+h) - f(x)]/h, for
h->0

We'll apply the principle to the given
polynomial:

lim {[(x+h)^2 + 6(x+h) + 10]-(x^2 + 6x +
10)}/h

The next step is to expand the
square:

lim {[(x^2+2xh + h^2) + 6x+6h + 10]-(x^2 + 6x +
10)}/h

We'll remove the brackets and combine and eliminate
like terms:

lim (2xh + h^2 +
6h)/h

We'll factorize by
h:

lim h(2x + h + 6)/h

We'll
simplify and we'll get:

lim (2x + h +
6)

We'll substitute h by 0 and we'll
get:

lim (2x + h + 6) = 2x +
6

So, the first derivative of the given function
is:

f'(x) = 2x +
6