What is the derivative of the this functon?
g(x) = -500/x, x cannot equal 0.
I know that in order to fnd the derivative I need to put the function into the equation for evaluating derivatives as limits.
lim as h -> 0 (f(x+h) - f(x))/h
I did this, but I am having difficulties simplifying it to evaluate the limit. Any help at all is appreciated!
Let K= -500
lim (K/(x+h) - K/x)/h
lim K (x-x-h)/x(x+h)h
= -K (h/x(x+h)h=lim -K 1/x(x+h)
= -k/x^2
= 500/x^2
Okay. So, 500/x^2 would be the simplified equation for the derivative, and from here I can figure that the limit is 5,000,000 as h approaches 0. Is this correct?
I too was able to come to the derivative of 500/x^2
To find the derivative of the function g(x) = -500/x, you can use the definition of the derivative:
lim as h -> 0 (f(x+h) - f(x))/h
Let's substitute the function g(x) into this equation:
lim as h -> 0 [(-500/(x + h)) - (-500/x)] / h
Now, let's simplify the expression in the numerator:
[(-500/(x + h)) - (-500/x)] / h
= [-500/(x + h) + 500/x] / h
= [-500x/(x(x + h)) + 500(x + h)/(x(x + h))] / h
= [-500x + 500(x + h)] / (x(x + h)) / h
= [-500x + 500x + 500h] / (x(x + h)) / h
= 500h / (x(x + h)) / h
Now, let's evaluate the limit by canceling out h:
lim as h -> 0 500 / (x(x + h))
Now, we can substitute h = 0 into the expression:
500 / (x(x + 0))
= 500 / (x^2)
So, the derivative of the function g(x) = -500/x is:
g'(x) = 500 / (x^2)