Algebra

se f(x)= 1/x. calcule (f(x+h)-f(x))/x

Step-by-step solution with explanation

Final Answer

Step-by-step solution

1

Write out f(x+h) and f(x)

We substitute into the function. Replace x with (x+h) to get f(x+h), and keep f(x) as 1/x.
2

Plug into the expression given

We replace f(x+h) and f(x) with what we found. Notice the denominator here is just x, not h — that is exactly what the problem states.
3

Subtract the fractions in the numerator

To subtract fractions, find a common denominator of x(x+h). Then combine the numerators: x minus (x+h) equals negative h.
4

Divide that result by x

Dividing by x is the same as multiplying by 1/x. So we multiply the denominator by x, giving x squared times (x+h).

Understanding this problem

Learning Insight

The expression (f(x+h) - f(x))/h is the classic difference quotient used in calculus to find a derivative. Here the denominator is x instead of h, which changes the result but uses the same fraction-subtraction algebra. Understanding how to combine fractions with unlike denominators is the core skill needed for both versions.

Quick Tip

When subtracting two fractions 1/(x+h) − 1/x, always cross-multiply the tops: the numerator becomes x − (x+h) = −h. This pattern appears constantly with rational functions.

Common Mistake

Students often forget to distribute the negative sign: they write x − x + h instead of x − (x+h) = x − x − h = −h, which flips the sign of the answer.