Use the Limit Definition to Find the Derivative x^3

Problem

d(x^3)/d(x)

Solution

1. Identify the limit definition of the derivative, which is given by the formula ƒ(x)^′=(lim_h→0)((ƒ*(x+h)−ƒ(x))/h)

2. Substitute the function ƒ(x)=x^3 into the limit definition.

(lim_h→0)(((x+h)^3−x^3)/h)

3. Expand the binomial (x+h)^3 using the binomial expansion formula (a+b)^3=a^3+3*a^2*b+3*a*b^2+b^3

(lim_h→0)((x^3+3*x^2*h+3*x*h^2+h^3−x^3)/h)

4. Simplify the numerator by canceling the x^3 and −x^3 terms.

(lim_h→0)((3*x^2*h+3*x*h^2+h^3)/h)

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

(lim_h→0)((h*(3*x^2+3*x*h+h^2))/h)

6. Divide the numerator and denominator by h

(lim_h→0)(3*x^2+3*x*h+h^2)

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

3*x^2+3*x(0)+(0)^2

3*x^2

Final Answer

d(x^3)/d(x)=3*x^2

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