How do you find a derivative?

To find the derivative of a given function, plug the function into the derivative formula, distribute all of the terms, and then cancel all possible terms to eliminate the formula's denominator. Once the formula is simplified, set h to approach zero to find the derivative.

1. Plug the function into the derivative formula

   The formula for the derivative of f(x) with respect to x is f'(x) = lim(h->0) ((f(x + h) - f(x)) / h). If the given function is f(x) = x^2 - 8x + 12, plug the function into the numerator of the formula. The result is f'(x) = lim(h->0) (((x + h)^2 - 8(x + h) + 12 - (x^2 - 8x - 12)) / h).

2. Distribute all of the terms

   Distribute all of the terms in the numerator, paying attention to any subtraction symbols. Once the function f(x) = x^2 -8x + 12 is plugged into the derivative formula and distributed, it becomes f'(x) = lim(h->0) ((x^2 + 2xh + h^2 - 8x - 8h + 12 - x^2 + 8x - 12) / h).

3. Eliminate the denominator

   Cancel out terms, and simplify the formula to eliminate the formula's denominator. The function f(x) = x^2 -8x + 12 is simplified into f'(x) = lim(h->0) 2x + h - 8.

4. Find the derivative

   Set h to approach zero. The result is the derivative of the given function. Thus, the derivative of f(x) = x^2 -8x + 12 is f'(x) = 2x - 8.