`d/(dx)(x^(5)cotx)=…………………. .`

Answer

To solve the problem \( \frac{d}{dx}(x^5 \cot x) \), we will use the product rule of differentiation. The product rule states that if you have two functions \( A \) and \( B \), then the derivative of their product is given by:

\[ \frac{d}{dx}(A \cdot B) = A \cdot \frac{dB}{dx} + B \cdot \frac{dA}{dx} \]

In this case, let:
- \( A = x^5 \)
- \( B = \cot x \)

Now, we will differentiate each part.

### Step 1: Differentiate \( A = x^5 \)

Using the power rule:
\[ \frac{dA}{dx} = 5x^{5-1} = 5x^4 \]

### Step 2: Differentiate \( B = \cot x \)

The derivative of \( \cot x \) is:
\[ \frac{dB}{dx} = -\csc^2 x \]

### Step 3: Apply the Product Rule

Now, we can apply the product rule:
\[ \frac{d}{dx}(x^5 \cot x) = A \cdot \frac{dB}{dx} + B \cdot \frac{dA}{dx} \]
Substituting the values we found:
\[ \frac{d}{dx}(x^5 \cot x) = x^5 \cdot (-\csc^2 x) + \cot x \cdot (5x^4) \]

### Step 4: Simplify the Expression

Now, we can simplify the expression:
\[ \frac{d}{dx}(x^5 \cot x) = -x^5 \csc^2 x + 5x^4 \cot x \]

Thus, the final result is:
\[ \frac{d}{dx}(x^5 \cot x) = 5x^4 \cot x - x^5 \csc^2 x \]