Use the first principle to determine the derivative of the function. f(x) = 5x^3 + 3x^2 - 2x + 15

According to the first principle of derivation, the
derivative of f(x) or f'(x)=
lim(h->0){[f(x+h)-f(x)]/h}

Here
f(x)=5x^3+3x^2-2x+15

f(x+h)=5(x+h)^3+3(x+h)^2-2(x+h)+15

=5*x^3+5*h^3+15*x^2*h+15*x*h^2+3*x^2+3*h^2+6*x*h-2x-2h+15

f(x+h)-f(x)=5*x^3+5h^3+15x^2*h+15x*h^2+3x^2+3h^2+6xh-2x-2h+15-5x^3+3x^2-2x+15

=5h^3+15x^2h+15xh^2+3h^2+6xh-2h

[f(x+h)-f(x)]/h=(5h^3+15x^2*h+15x*h^2+3h^2+6xh-2h)/h

=5h^2+15x^2+15xh+3h+6x-2

lim(h->0)[5h^2+15x^2+15xh+3h+6x-2]

=15x^2+6x-2

Therefore
(5x^3+3x^2-2x+15)' by the first principle is
15x^2+6x-2.