Simplify ((x+h)^-2-x^-2)/h

Problem

((x+h)(−2)−x(−2))/h

Solution

1. Rewrite the negative exponents as fractions to make the expression easier to manipulate.

(1/((x+h)2)−1/(x2))/h

2. Find a common denominator for the terms in the numerator, which is x2*(x+h)2

(x2−(x+h)2)/(x2*(x+h)2)/h

3. Expand the squared binomial (x+h)2 in the numerator.

(x2−(x2+2*x*h+h2))/(x2*(x+h)2)/h

4. Distribute the negative sign and simplify the numerator of the complex fraction.

(x2−x2−2*x*h−h2)/(x2*(x+h)2)/h

(−2*x*h−h2)/(x2*(x+h)2)/h

5. Divide by h by multiplying the denominator of the top fraction by h

(−2*x*h−h2)/(h⋅x2*(x+h)2)

6. Factor out h from the numerator to prepare for cancellation.

(h*(−2*x−h))/(h⋅x2*(x+h)2)

7. Cancel the common factor h from the numerator and denominator.

(−2*x−h)/(x2*(x+h)2)

Final Answer

((x+h)(−2)−x(−2))/h=(−2*x−h)/(x2*(x+h)2)

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