Use the Limit Definition to Find the Derivative f(x)=x^3

Problem

ƒ(x)=x3

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)=x3 into the definition.

d(ƒ(x))/d(x)=(lim_h→0)(((x+h)3−x3)/h)

3. Expand the binomial (x+h)3 using the binomial theorem or algebraic expansion.

d(ƒ(x))/d(x)=(lim_h→0)((x3+3*x2*h+3*x*h2+h3−x3)/h)

4. Simplify the numerator by canceling the x3 and −x3 terms.

d(ƒ(x))/d(x)=(lim_h→0)((3*x2*h+3*x*h2+h3)/h)

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

d(ƒ(x))/d(x)=(lim_h→0)((h*(3*x2+3*x*h+h2))/h)

6. Divide by h to eliminate the indeterminate form.

d(ƒ(x))/d(x)=(lim_h→0)(3*x2+3*x*h+h2)

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

d(ƒ(x))/d(x)=3*x2+3*x(0)+(0)2

Final Answer

d(x3)/d(x)=3*x2

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