To find the derivative of the function f(x) = 1/x, we can use the rules of differentiation applied to polynomial and rational functions. Let’s begin by rewriting the function in a more manageable form.
We know that:
f(x) = 1/x can be rewritten as f(x) = x-1.
Now, we can use the power rule, which states that if f(x) = xn, then f'(x) = n imes xn-1. Applying this rule here gives:
f'(x) = -1 imes x-1-1 = -1 imes x-2.
Thus, we can write the derivative as:
f'(x) = -1/x2.
This means that the derivative of 1/x is -1/x2. This result indicates how the function 1/x changes with respect to x.
To summarize:
- The original function is 1/x or x-1.
- The derivative of 1/x is -1/x2.
This derivative tells us that as x increases, the function 1/x decreases at a rate proportional to the square of x.