Q. \[ \frac{d}{dx}\left(\frac{6}{x}\right) \]

Q: What is the derivative of \frac{6}{x} ?

Rewrite \frac{6}{x}=6x^{-1} . Then \frac{d}{dx}\left(6x^{-1}\right)=6(-1)x^{-2}=-\frac{6}{x^2} .

Q: How do you use the power rule on 6x^{-1} ?

The power rule says \frac{d}{dx}(x^n)=nx^{n-1} . With n=-1 : \frac{d}{dx}(6x^{-1})=6(-1)x^{-2} .

Q: What is the derivative using quotient rule for \frac{6}{x} ?

Let f=\frac{6}{x}=\frac{6}{x} . Quotient rule: \frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^2} . Here u=6, u'=0, v=x, v'=1 . Result -\frac{6}{x^2} .

Q: What derivative rules apply to constants divided by x ?

Treat constants: \frac{d}{dx}\left(\frac{c}{x}\right)=c\frac{d}{dx}(x^{-1})=-cx^{-2}=-\frac{c}{x^2} . For c=6 , it is -\frac{6}{x^2} .

Q: What is the domain of \frac{d}{dx}\left(\frac{6}{x}\right) ?

The original function \frac{6}{x} is defined for x\neq 0 . The derivative -\frac{6}{x^2} is also defined for x\neq 0 .

Q: Can we express -\frac{6}{x^2} in powers of x ?

Yes. Since \frac{1}{x^2}=x^{-2} , then -\frac{6}{x^2}=-6x^{-2} . Both forms are equivalent.

Answer

Rewrite \( \frac{6}{x} = 6x^{-1} \). Differentiate using the power rule \( \frac{d}{dx}x^{n} = nx^{n-1} \):

\[
\frac{d}{dx}\left(6x^{-1}\right)=6(-1)x^{-2}=-6x^{-2}.
\]

Convert back to a fraction:

\[
-6x^{-2}=-\frac{6}{x^{2}}.
\]

Final result: \(-\frac{6}{x^{2}}\).

Detailed Explanation

We want to find the derivative of the function \( \frac{6}{x} \).

Step 1: Rewrite the function using exponents

Recall that dividing by \(x\) is the same as multiplying by \(x^{-1}\). So we rewrite:

\[
\frac{6}{x} = 6x^{-1}.
\]

Step 2: Use the power rule

The power rule says: if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\).

Here, \(f(x) = 6x^{-1}\). Treat the \(6\) as a constant factor outside the derivative.

Step 3: Differentiate

Differentiate \(6x^{-1}\):

\[
\frac{d}{dx}\left(6x^{-1}\right) = 6 \cdot (-1)x^{-1-1}.
\]

Step 4: Simplify the exponent

Compute the exponent \( -1 – 1 = -2 \):

\[
6 \cdot (-1)x^{-2} = -6x^{-2}.
\]

Step 5: Rewrite in fraction form (optional)

Since \(x^{-2} = \frac{1}{x^2}\), we get:

\[
-6x^{-2} = -\frac{6}{x^2}.
\]

Final Answer

\[
\frac{d}{dx}\left(\frac{6}{x}\right) = -\frac{6}{x^2}.
\]

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Calculus FAQ

What is the derivative of \( \frac{6}{x} \)?

Rewrite \( \frac{6}{x}=6x^{-1} \). Then \( \frac{d}{dx}\left(6x^{-1}\right)=6(-1)x^{-2}=-\frac{6}{x^2} \).

How do you use the power rule on \( 6x^{-1} \)?

The power rule says \( \frac{d}{dx}(x^n)=nx^{n-1} \). With \( n=-1 \): \( \frac{d}{dx}(6x^{-1})=6(-1)x^{-2} \).

What is the derivative using quotient rule for \( \frac{6}{x} \)?

Let \( f=\frac{6}{x}=\frac{6}{x} \). Quotient rule: \( \frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^2} \). Here \( u=6, u'=0, v=x, v'=1 \). Result \( -\frac{6}{x^2} \).

What derivative rules apply to constants divided by \( x \)?

Treat constants: \( \frac{d}{dx}\left(\frac{c}{x}\right)=c\frac{d}{dx}(x^{-1})=-cx^{-2}=-\frac{c}{x^2} \). For \( c=6 \), it is \( -\frac{6}{x^2} \).

What is the domain of \( \frac{d}{dx}\left(\frac{6}{x}\right) \)?

The original function \( \frac{6}{x} \) is defined for \( x\neq 0 \). The derivative \( -\frac{6}{x^2} \) is also defined for \( x\neq 0 \).

Can we express \( -\frac{6}{x^2} \) in powers of \( x \)?

Yes. Since \( \frac{1}{x^2}=x^{-2} \), then \( -\frac{6}{x^2}=-6x^{-2} \). Both forms are equivalent.

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