Which of the following corresponds to the derivative of f(x)=1/x+2 at x = 2, using the alternate definition of a derivative, reduced to its simplest form before taking the limit?

Answer 1

We can use the alternate definition of a derivative to find the derivative of f(x) = 1/x + 2 at x = 2.

The alternate definition of a derivative is given by:

f'(x) = lim(h→0) [f(x + h) - f(x)] / h

Plugging in the given function f(x) = 1/x + 2, we can calculate the derivative:

f'(x) = lim(h→0) [f(x + h) - f(x)] / h
f'(x) = lim(h→0) [(1/(x+h) + 2) - (1/x + 2)] / h
f'(x) = lim(h→0) [(1/(x+h) - 1/x)] / h

To simplify this expression, we need to find a common denominator for the two fractions. Multiplying the first fraction by (x/x) and the second fraction by ((x+h)/(x+h)), we get:

f'(x) = lim(h→0) [(x - (x+h)) / (x(x+h))] / h
f'(x) = lim(h→0) [(x - x - h) / (x(x+h))] / h
f'(x) = lim(h→0) [-h / (x(x+h))] / h
f'(x) = lim(h→0) -1 / (x(x+h))

Taking the limit as h approaches 0, we have:

f'(x) = -1 / (x(x+0))
f'(x) = -1 / (x^2)

Now, we need to find the derivative at x = 2:

f'(2) = -1 / (2^2)
f'(2) = -1 / 4

Therefore, the derivative of f(x) = 1/x + 2 at x = 2, using the alternate definition of a derivative and reduced to its simplest form before taking the limit, is -1/4.