Q. \[ \frac{d}{dx}\left(x^{3}\right) \]
Answer
To find the derivative of \(x^3\), use the power rule: \(\frac{d}{dx}\big(x^n\big)=n x^{n-1}\).
\[
\frac{d}{dx}\left(x^3\right)=3x^{3-1}=3x^2
\]
Final result: \(3x^2\)
Detailed Explanation
We want to find the derivative of the function
\[
f(x)=x^3.
\]
Step 1: Use the power rule.
The power rule says that for any real number \(n\),
\[
\frac{d}{dx}\left(x^n\right)=n x^{n-1}.
\]
Step 2: Identify \(n\).
Here, \(x^3\) has exponent \(n=3\).
Step 3: Apply the power rule.
\[
\frac{d}{dx}\left(x^3\right)=3x^{3-1}.
\]
Step 4: Simplify the exponent.
\[
3-1=2,
\]
so the derivative becomes
\[
\frac{d}{dx}\left(x^3\right)=3x^2.
\]
Final answer:
\[
\boxed{\frac{d}{dx}\left(x^3\right)=3x^2.}
\]
See full solution
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Calculus FAQ
What is \( \frac{d}{dx}\left(x^3\right) \)?
\( \frac{d}{dx}\left(x^3\right)=3x^2 \).
How do you use the power rule for \(x^n\)?
If \(y=x^n\), then \( \frac{dy}{dx}=nx^{n-1} \). For \(x^3\), \(n=3\), so \(3x^{2}\).
What is the derivative of \(x^3+5\)?
\( \frac{d}{dx}\left(x^3+5\right)=3x^2+0=3x^2 \).
What is the derivative of \(4x^3\)?
Use the constant multiple rule: \( \frac{d}{dx}\left(4x^3\right)=4\cdot 3x^2=12x^2 \).
What is the derivative of \(x^3\) using the limit definition?
From \( \lim_{h\to 0}\frac{(x+h)^3-x^3}{h} \), expand to get \(3x^2+3xh+h^2\), and the limit is \(3x^2\).
What is the second derivative of \(x^3\)?
First derivative is \(3x^2\). Differentiate again: \( \frac{d}{dx}(3x^2)=6x \).
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