What is the derivative of x^3?
Understand the Problem
The question is asking for the derivative of the function x^3. The derivative is a fundamental concept in calculus that represents the rate of change of the function with respect to x.
Answer
3x^2
Answer for screen readers
The derivative of $x^3$ is $3x^2$
Steps to Solve
- Apply the power rule
To find the derivative of a function of the form $x^n$, use the power rule: $$\frac{d}{dx}[x^n] = nx^{n-1}$$
- Substitute the exponent
In this problem, the exponent $n$ is 3. Substitute $n = 3$ into the power rule equation: $$\frac{d}{dx}[x^3] = 3x^{3-1}$$
- Simplify the expression
Subtract 1 from the exponent: $$3x^{3-1} = 3x^2$$
The derivative of $x^3$ is $3x^2$
More Information
The derivative tells you how the function $x^3$ changes as x changes. For example, when x increases by a tiny amount, the rate at which $x^3$ increases is about 3 times the square of x.
Tips
A common mistake is forgetting to subtract 1 from the exponent when applying the power rule. Always remember the correct form of the rule: $\frac{d}{dx}[x^n] = nx^{n-1}$
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