`(d)/(dx)[log((x^(n))/(e^(x)))]=`

Options

A. `(n)/(x)+1`
B. `(1)/(x)+n`
C. `(n)/(x)-1`
D. `(1)/(x)-1`

Answer

To differentiate the function \( \log\left(\frac{x^n}{e^x}\right) \), we can follow these steps:

### Step 1: Apply the logarithmic property
Using the property of logarithms that states \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \), we can rewrite the expression:
\[ \log\left(\frac{x^n}{e^x}\right) = \log(x^n) - \log(e^x) \]

### Step 2: Simplify using logarithmic identities
Next, we can use the property of logarithms that states \( \log(a^m) = m \log(a) \):
\[ \log(x^n) = n \log(x) \quad \text{and} \quad \log(e^x) = x \]
Thus, we can rewrite the expression as:
\[ n \log(x) - x \]

### Step 3: Differentiate the expression
Now, we differentiate the expression \( n \log(x) - x \) with respect to \( x \):
\[ \frac{d}{dx}[n \log(x) - x] \]
Using the derivative rules, we have:
- The derivative of \( n \log(x) \) is \( n \cdot \frac{1}{x} \) (since \( n \) is a constant).
- The derivative of \( -x \) is \( -1 \).

Putting it all together, we get:
\[ \frac{d}{dx}[n \log(x) - x] = \frac{n}{x} - 1 \]

### Final Answer
Thus, the derivative of \( \log\left(\frac{x^n}{e^x}\right) \) is:
\[ \frac{n}{x} - 1 \]

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