Find the derivative of the following functions :
`3cotx+5cosecx`

Answer

To find the derivative of the function \( f(x) = 3\cot x + 5\csc x \), we will follow these steps:

### Step 1: Write the derivative expression
We need to find the derivative of the function with respect to \( x \):
\[ \frac{d}{dx}(3\cot x + 5\csc x) \]

### Step 2: Apply the derivative operator
Using the linearity of differentiation, we can separate the terms:
\[ \frac{d}{dx}(3\cot x) + \frac{d}{dx}(5\csc x) \]

### Step 3: Factor out constants
Since 3 and 5 are constants, we can factor them out:
\[ 3 \frac{d}{dx}(\cot x) + 5 \frac{d}{dx}(\csc x) \]

### Step 4: Differentiate \( \cot x \) and \( \csc x \)
Now, we will use the known derivatives:
- The derivative of \( \cot x \) is \( -\csc^2 x \).
- The derivative of \( \csc x \) is \( -\csc x \cot x \).

Substituting these derivatives into our expression gives:
\[ 3(-\csc^2 x) + 5(-\csc x \cot x) \]

### Step 5: Simplify the expression
This simplifies to:
\[ -3\csc^2 x - 5\csc x \cot x \]

### Step 6: Factor out the negative sign
We can factor out a negative sign:
\[ -(3\csc^2 x + 5\csc x \cot x) \]

### Final Answer
Thus, the derivative of the function \( 3\cot x + 5\csc x \) is:
\[ -(3\csc^2 x + 5\csc x \cot x) \]

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