Find f'(a).
f(x)= (x^2 + 1)/(x - 2)
So,
[[(a + h)^2 + 1]/[(a + h) - 2]] - [(a^2 + 1)/a - 2] and the whole thing divided by h
I tried simplifying it and ended up with [a + h/-2] - (a-2), which is incorrect. I used wolfram alpha and got (a^2 - 4a - 1)/(a - 2)^2 but I do not know how to get this. Please help.
To find the derivative, f'(a), of the function f(x) = (x^2 + 1)/(x - 2), we need to use the definition of the derivative. The derivative of a function measures the rate at which the function changes as its input changes.
Let's start by applying the definition of the derivative to find f'(a):
f'(a) = lim(h -> 0) [(f(a + h) - f(a)) / h]
Plugging in the function f(x) = (x^2 + 1)/(x - 2), we have:
f'(a) = lim(h -> 0) [((a + h)^2 + 1) / (a + h - 2) - ((a^2 + 1) / (a - 2))] / h
Now, let's simplify the numerator:
[(a + h)^2 + 1] / (a + h - 2) - (a^2 + 1) / (a - 2)
To combine these two fractions, we need to find a common denominator:
[(a + h)^2 + 1] * (a - 2) - (a^2 + 1) * (a + h - 2) / (a + h - 2)(a - 2)
Expanding and simplifying this expression, we get:
[(a^2 - 4a - 1) + 2ah + h^2] - [(-a^2 + 3a - 1) + (a^2 - 4a + 1)h] / (a + h - 2)(a - 2)
Combining like terms, we have:
(a^2 - 4a - 1 + 2ah + h^2 + a^2 - 3a + 1 - a^2 + 4a - ah) / (a + h - 2)(a - 2)
Simplifying further:
(- 4ah + 2h^2 + 1) / (a + h - 2)(a - 2)
Now we can take the limit as h approaches 0:
lim(h -> 0) (- 4ah + 2h^2 + 1) / (a + h - 2)(a - 2)
To evaluate this limit, we substitute h = 0 into the expression:
(- 4a(0) + 2(0)^2 + 1) / (a + 0 - 2)(a - 2)
= (1) / (a - 2)(a - 2)
= 1 / (a - 2)^2
Finally, we have the derivative:
f'(a) = 1 / (a - 2)^2
So, f'(a) is equal to 1 divided by the square of the difference between a and 2, or (a - 2)^2.