Q. \[ \frac{d}{dx}\left(x^{2}\right) \]

Q: How do you use the power rule to differentiate x^2?

Use \frac{d}{dx}(x^n)=nx^{n-1} . With n=2, get 2x^{1}=2x.

Q: What is the derivative of 2x^2?

Constant multiple rule: \frac{d}{dx}(2x^2)=2\cdot \frac{d}{dx}(x^2)=2(2x)=4x .

Q: Find the derivative of (x^2)^3.

Rewrite (x^2)^3=x^6 . Then \frac{d}{dx}(x^6)=6x^5 .

Q: What is the derivative of x^{-1}\cdot x^2?

Simplify first: x^{-1}\cdot x^2=x^{1}=x. Then \frac{d}{dx}(x)=1 .

Answer

To differentiate the function \(x^2\), use the power rule: \( \frac{d}{dx}(x^n)=n x^{n-1}\).

\[
\frac{d}{dx}(x^2)=2x^{2-1}=2x
\]

Final result: \(2x\)

Detailed Explanation

We want to find the derivative of the function \(f(x) = x^2\).

Step 1: Write the function clearly

Let

\[
f(x) = x^2
\]

The derivative we want is \(f'(x)\).

Step 2: Use the power rule

The power rule says: if \(f(x) = x^n\), then

\[
\frac{d}{dx}\left(x^n\right) = n x^{n-1}
\]

Here, \(n = 2\).

Step 3: Apply the power rule

Substitute \(n = 2\) into the formula:

\[
\frac{d}{dx}\left(x^2\right) = 2x^{2-1}
\]

Step 4: Simplify the exponent

Compute the exponent \(2-1\):

\[
2x^{2-1} = 2x^1
\]

Step 5: Write the final simplified result

Since \(x^1 = x\), we get:

\[
\frac{d}{dx}\left(x^2\right) = 2x
\]

Final Answer:

\[
\frac{d}{dx}\left(x^2\right) = 2x
\]

See full solution

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Calculus FAQ

What is \( \frac{d}{dx}(x^2) \)?

\( \frac{d}{dx}(x^2)=2x \).

How do you use the power rule to differentiate \(x^2\)?

Use \( \frac{d}{dx}(x^n)=nx^{n-1} \). With \(n=2\), get \(2x^{1}=2x\).

What is the derivative of \(2x^2\)?

Constant multiple rule: \( \frac{d}{dx}(2x^2)=2\cdot \frac{d}{dx}(x^2)=2(2x)=4x \).

What is the derivative of \(x^2+3\)?

\( \frac{d}{dx}(x^2+3)=\frac{d}{dx}(x^2)+\frac{d}{dx}(3)=2x+0=2x \).

Find the derivative of \((x^2)^3\).

Rewrite \( (x^2)^3=x^6 \). Then \( \frac{d}{dx}(x^6)=6x^5 \).

What is the derivative of \(x^{-1}\cdot x^2\)?

Simplify first: \(x^{-1}\cdot x^2=x^{1}=x\). Then \( \frac{d}{dx}(x)=1 \).

Use this for the derivative of x².
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