Use the Limit Definition to Find the Derivative f(x)=2/x

Problem

ƒ(x)=2/x

Solution

1. State the limit definition of the derivative, which is the limit of the difference quotient as h approaches zero.

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

2. Substitute the function ƒ(x)=2/x into the definition.

d(ƒ(x))/d(x)=(lim_h→0)((2/(x+h)−2/x)/h)

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

d(ƒ(x))/d(x)=(lim_h→0)((2*x−2*(x+h))/(x*(x+h))/h)

4. Simplify the numerator by distributing the constant and subtracting the terms.

d(ƒ(x))/d(x)=(lim_h→0)((2*x−2*x−2*h)/(x*(x+h))/h)

d(ƒ(x))/d(x)=(lim_h→0)((−2*h)/(x*(x+h))/h)

5. Divide by h by multiplying the numerator fraction by the reciprocal of the denominator.

d(ƒ(x))/d(x)=(lim_h→0)((−2*h)/(h⋅x*(x+h)))

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

d(ƒ(x))/d(x)=(lim_h→0)((−2)/(x*(x+h)))

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

d(ƒ(x))/d(x)=(−2)/(x*(x+0))

d(ƒ(x))/d(x)=−2/(x2)

Final Answer

d()/d(x)2/x=−2/(x2)

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