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The derivative of $x^3$ .

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Lesson Parent:

Differentiation II: The Gory Details of Calculating Derivatives

Problem:

What is the derivative of $x^3$ ?

Answer:

$f\,'(x)=3\,x^2$

Solution:

Recall that the derivative is defined as

<Sign in to see all the formulas>

For the function $x^3$ we have

<Sign in to see all the formulas>

Recall that we know <Sign in to see all the formulas>, <Sign in to see all the formulas>, and <Sign in to see all the formulas>. Because

the limit of a function times a constant is the function's limit times the constant,
$3x^2$ and $3x$ are constants here (remember $x$ is some number we can fill in later)
and the limit of the sum of two functions is the sum of their limits

we have

<Sign in to see all the formulas>