Question 515569
The difference quotient of a function f(x) is:

{{{(f(x + h) - f(x))/(h)}}}

It measures the slope of the line that passes through the graph y = f(x) at x and x + h, where h is typically a very small number.  If we are able to take the limit as h approaches 0, the resulting number is the derivative f'(x).

In this case, {{{f(x) = x^4}}}, so

{{{f(x + h) = (x + h)^4}}}
{{{f(x + h) = x^4 + 4(x^3)h + 6(x^2)(h^2) + 4x(h^3) + h^4 }}} (multiplying out {{{(x + h)^4}}}, with the intermediate steps skipped --- write out the product and distribute carefully)

Substituting into the formula for the difference quotient, we get:

{{{(f(x + h) - f(x))/(h)}}}
{{{(x^4 + 4(x^3)h + 6(x^2)(h^2) + 4x(h^3) + h^4 - x^4)/(h)}}} (substituting)
{{{(4(x^3)h + 6(x^2)(h^2) + 4x(h^3) + h^4)/(h)}}} (combining like terms)
{{{4 x^3 + 6(x^2)h + 4x(h^2) + h^3 }}} (canceling h)

Notice that as h goes to 0, all of the terms of the difference quotient go to 0 except for the first, leaving {{{4x^3}}} as the derivative of {{{x^4}}}.