Use the Limit Definition to Find the Derivative 1/x

Problem

d()/d(x)1/x

Solution

1. State the limit definition of the derivative for the function ƒ(x)=1/x

d(ƒ)/d(x)=(lim_h→0)((ƒ*(x+h)−ƒ(x))/h)

2. Substitute the function into the definition.

d()/d(x)1/x=(lim_h→0)((1/(x+h)−1/x)/h)

3. Find a common denominator for the terms in the numerator to combine the fractions.

d()/d(x)1/x=(lim_h→0)((x−(x+h))/(x*(x+h))/h)

4. Simplify the numerator by distributing the negative sign and subtracting x

d()/d(x)1/x=(lim_h→0)((−h)/(x*(x+h))/h)

5. Divide by h by multiplying the numerator's fraction by 1/h

d()/d(x)1/x=(lim_h→0)((−h)/(h⋅x*(x+h)))

6. Cancel the common factor of h from the numerator and denominator.

d()/d(x)1/x=(lim_h→0)((−1)/(x*(x+h)))

7. Evaluate the limit by substituting h=0 into the expression.

d()/d(x)1/x=(−1)/(x*(x+0))

8. Simplify the final expression.

d()/d(x)1/x=−1/(x2)

Final Answer

d()/d(x)1/x=−1/(x2)

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