To find the derivative of the function f(x) = 2x / (x + 7) using the alternative formula for the derivative, we start by recalling the definition of the derivative:
f'(x) = lim (h -> 0) [f(x + h) – f(x)] / h
First, we compute f(x + h):
f(x + h) = 2(x + h) / ((x + h) + 7) = 2(x + h) / (x + h + 7)
Now we plug f(x + h) and f(x) into the derivative formula:
f'(x) = lim (h -> 0) [2(x + h) / (x + h + 7) – 2x / (x + 7)] / h
Next, we simplify the expression inside the limit:
= lim (h -> 0) [2(x + h)(x + 7) – 2x(x + h + 7)] / [(x + h + 7)(x + 7)h]
Distributing the terms yields:
= lim (h -> 0) [2(x^2 + 7h + xh + h^2) – 2(x^2 + 7x)] / [(x + h + 7)(x + 7)h]
Now simplify further:
= lim (h -> 0) [2xh + 14h + 2h^2] / [(x + h + 7)(x + 7)h]
Factor out h from the numerator:
= lim (h -> 0) [h(2x + 14 + 2h)] / [(x + h + 7)(x + 7)h]
We can then cancel h:
= lim (h -> 0) [2x + 14 + 2h] / [(x + h + 7)(x + 7)]
Finally, substitute h = 0:
= [2x + 14] / [(x + 7)(x + 7)]
Therefore, the derivative of the function f(x) = 2x / (x + 7) is:
f'(x) = (2x + 14) / (x + 7)^2