Use the Limit Definition to Find the Derivative x^2

Problem

d(x2)/d(x)

Solution

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

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

2. Substitute the function ƒ(x)=x2 into the definition.

d(x2)/d(x)=(lim_h→0)(((x+h)2−x2)/h)

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

d(x2)/d(x)=(lim_h→0)((x2+2*x*h+h2−x2)/h)

4. Simplify the numerator by canceling the x2 and −x2 terms.

d(x2)/d(x)=(lim_h→0)((2*x*h+h2)/h)

5. Factor out h from the numerator to prepare for division.

d(x2)/d(x)=(lim_h→0)((h*(2*x+h))/h)

6. Divide the numerator and denominator by h

d(x2)/d(x)=(lim_h→0)(2*x+h)

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

d(x2)/d(x)=2*x+0

Final Answer

d(x2)/d(x)=2*x

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