Okay, I know that the derivative functon of the function -500/x is 500/x^2, but I am having a hard time getting to this result using limit notation. Could someone show or explain the steps used in limit notation to get the derivative of 500/x^2?
Of course! To find the derivative of the function -500/x using limit notation, we can use the definition of the derivative. The derivative of a function f(x) at a specific point can be found using the following formula:
f'(x) = lim(h→0) [f(x+h) - f(x)] / h
Now, let's apply this formula to -500/x.
Step 1: Start with the definition of the derivative:
f'(x) = lim(h→0) [-500/(x+h) - (-500/x)] / h
Step 2: Simplify the expression inside the limit notation:
f'(x) = lim(h→0) [-500x + 500(x + h)] / (x(x + h)) / h
Step 3: Distribute and simplify the numerator:
f'(x) = lim(h→0) [-500x + 500x + 500h] / (x(x + h)) / h
Step 4: Further simplify:
f'(x) = lim(h→0) [500h] / (x(x + h)) / h
Step 5: Cancel out h in the numerator and denominator:
f'(x) = lim(h→0) 500 / (x(x + h))
Step 6: Now, let's evaluate the limit as h approaches zero:
f'(x) = 500 / (x(x + 0))
Step 7: Simplify further:
f'(x) = 500 / (x^2)
So, the derivative function of -500/x using limit notation is 500/x^2.