Show the steps to get the derivative function of -500/x by evaluating the limit. By limit, they are referring to the equation, lim h -> 0 (f(x + h)-f(x))/h
lim x>0 (-500/(x+h) - 500/x )/h
= lim -500(x-(x+h))/h(x*(x+h)
= lim -500 (-h/h(x*(x+h))
= lim +500 (1/x(x+h))
= 500/x^2
Your teacher is just too easy.
To find the derivative function of -500/x using the limit definition, we need to evaluate the following limit:
lim h → 0 [f(x + h) - f(x)] / h
Here, f(x) represents the function -500/x. Let's go through the steps to get the derivative function:
Step 1: Start with the given function f(x) = -500/x.
Step 2: Substitute the value of f(x + h) and f(x) into the equation.
[f(x + h) - f(x)] / h = [(-500 / (x + h)) - (-500 / x)] / h
Step 3: Simplify the expression inside the brackets.
= [-500(x) - (-500(x + h))] / (hx(x + h))
= [-500x + 500(x + h)] / (hx(x + h))
= [-500x + 500x + 500h] / (hx(x + h))
= (500h) / (hx(x + h))
Step 4: Cancel out the h term in the numerator and denominator.
= 500 / (x(x + h))
Step 5: Take the limit as h approaches 0.
lim h → 0 (500 / (x(x + h)))
At this point, we cannot evaluate the limit directly since there is still an 'h' term in the denominator. However, we can rewrite the limit to make it easier to evaluate.
Step 6: Expand the denominator using the distributive property.
= 500 / (x^2 + xh)
Step 7: As h approaches 0, xh approaches 0 as well, so the term xh in the denominator will vanish.
= 500 / (x^2)
Step 8: Simplify the expression.
= 500x^(-2)
Therefore, the derivative function of -500/x, obtained by evaluating the limit, is 500x^(-2).