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

Options

A. `1-csc^(2)x`
B. `-1-csc^(2)x`
C. `x^(-1)-csc^(2)x`
D. `x^(-1)+csc^(2)x`

Answer

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

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

### Step 2: Simplify the logarithm of the exponential
Using the property of logarithms that states \( \log(a^b) = b \cdot \log(a) \), we can simplify further:
\[ \log(e^{\cot x}) = \cot x \cdot \log(e) \]
Since \( \log(e) = 1 \), we have:
\[ \log(e^{\cot x}) = \cot x \]

### Step 3: Rewrite the expression
Now, substituting back, we get:
\[ \log\left(\frac{x}{e^{\cot x}}\right) = \log(x) - \cot x \]

### Step 4: Differentiate the expression
Now we differentiate the expression:
\[ \frac{d}{dx}[\log(x) - \cot(x)] \]
Using the derivative formulas:
- The derivative of \( \log(x) \) is \( \frac{1}{x} \).
- The derivative of \( \cot(x) \) is \( -\csc^2(x) \).

Thus, we have:
\[ \frac{d}{dx}[\log(x)] - \frac{d}{dx}[\cot(x)] = \frac{1}{x} - (-\csc^2(x)) \]
This simplifies to:
\[ \frac{1}{x} + \csc^2(x) \]

### Final Answer
So, the final answer is:
\[ \frac{1}{x} + \csc^2(x) \]

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