Could someone please help me with this?
Calculate the instantaneous rate of change of f <note: f(x)=(x-6)/(x+6)> at x=6.
why the <note>? Just say what f(x) is. You can use the quotient rule if you want, but note that
f(x) = (x-6)/(x+6) = 1 - 12/(x+6)
so,
f'(x) = 12/(x+6)^2
f'(6) = 12/12^2 = 1/12
OH okay thank you so much!
Of course! I can help you with that. To calculate the instantaneous rate of change of a function at a specific point, we need to find the derivative of the function and evaluate it at that point.
In this case, you have the function f(x) = (x - 6)/(x + 6) and you want to find the instantaneous rate of change at x = 6.
To find the derivative of f(x), we will use the quotient rule. The quotient rule states that if you have a function in the form f(x) = g(x)/h(x), then the derivative of f(x) is given by the formula:
f'(x) = (g'(x)*h(x) - g(x)*h'(x))/(h(x))^2
Let's differentiate f(x) using the quotient rule:
f'(x) = [(1)*(x + 6) - (x - 6)*(1)] / (x + 6)^2
= (x + 6 - x + 6) / (x + 6)^2
= 12 / (x + 6)^2
Now that we have the derivative of f(x), we can evaluate it at x = 6 to find the instantaneous rate of change:
f'(6) = 12 / (6 + 6)^2
= 12 / (12)^2
= 12 / 144
= 1/12
Therefore, the instantaneous rate of change of f(x) at x = 6 is 1/12.
I hope this helps! Let me know if you have any further questions.