Find the derivative of the following w.e.t. x :
`3/(root(3)(x))-5/cosx+log_(3)x+6/sinx-(2tanx)/secx+7`

Answer

To find the derivative of the function

\[ f(x) = \frac{3}{\sqrt[3]{x}} - \frac{5}{\cos x} + \log_3 x + \frac{6}{\sin x} - \frac{2 \tan x}{\sec x} + 7, \]

we will differentiate each term step by step.

### Step 1: Rewrite the function in a more manageable form
We can rewrite the function using exponent and logarithmic properties:

\[ f(x) = 3x^{-1/3} - 5\sec x + \log_3 x + 6\csc x - 2\tan x \cos x + 7. \]

### Step 2: Differentiate each term
Now, we will differentiate each term separately.

1. **Derivative of \(3x^{-1/3}\)**:
\[ \frac{d}{dx}(3x^{-1/3}) = 3 \cdot (-\frac{1}{3})x^{-4/3} = -x^{-4/3}. \]

2. **Derivative of \(-5\sec x\)**:
\[ \frac{d}{dx}(-5\sec x) = -5\sec x \tan x. \]

3. **Derivative of \(\log_3 x\)**:
Using the change of base formula:
\[ \frac{d}{dx}(\log_3 x) = \frac{1}{x \ln 3}. \]

4. **Derivative of \(6\csc x\)**:
\[ \frac{d}{dx}(6\csc x) = 6(-\csc x \cot x) = -6\csc x \cot x. \]

5. **Derivative of \(-2\tan x \cos x\)**:
Using the product rule:
\[ \frac{d}{dx}(-2\tan x \cos x) = -2\left(\sec^2 x \cos x + \tan x (-\sin x)\right) = -2\sec^2 x \cos x + 2\tan x \sin x. \]

6. **Derivative of the constant \(7\)**:
\[ \frac{d}{dx}(7) = 0. \]

### Step 3: Combine all derivatives
Now, we combine all the derivatives we found:

\[ f'(x) = -x^{-4/3} - 5\sec x \tan x + \frac{1}{x \ln 3} - 6\csc x \cot x - 2\sec^2 x \cos x + 2\tan x \sin x. \]

### Final Answer
Thus, the derivative of the given function is:

\[ f'(x) = -\frac{1}{x^{4/3}} - 5\sec x \tan x + \frac{1}{x \ln 3} - 6\csc x \cot x - 2\sec^2 x \cos x + 2\tan x \sin x. \]