Find the derivative of the following w.e.t. x :
`log(1/sqrtx)+5x^(a)-3a^(x)+root(3)(x^(2))+6root(4)(x^(-3))`.

Answer

To find the derivative of the function

\[ y = \log\left(\frac{1}{\sqrt{x}}\right) + 5x^a - 3a^x + \sqrt[3]{x^2} + 6\sqrt[4]{x^{-3}}, \]

we will differentiate each term step by step.

### Step 1: Simplify the logarithmic term

The first term can be simplified:

\[ \log\left(\frac{1}{\sqrt{x}}\right) = \log(x^{-1/2}) = -\frac{1}{2} \log(x). \]

So, we rewrite \(y\):

\[ y = -\frac{1}{2} \log(x) + 5x^a - 3a^x + x^{2/3} + 6x^{-3/4}. \]

### Step 2: Differentiate each term

Now we differentiate each term with respect to \(x\):

1. For \(-\frac{1}{2} \log(x)\):
\[ \frac{d}{dx}\left(-\frac{1}{2} \log(x)\right) = -\frac{1}{2} \cdot \frac{1}{x} = -\frac{1}{2x}. \]

2. For \(5x^a\):
\[ \frac{d}{dx}(5x^a) = 5a x^{a-1}. \]

3. For \(-3a^x\):
Using the formula \(\frac{d}{dx}(a^x) = a^x \ln(a)\):
\[ \frac{d}{dx}(-3a^x) = -3a^x \ln(a). \]

4. For \(x^{2/3}\):
\[ \frac{d}{dx}(x^{2/3}) = \frac{2}{3} x^{-1/3}. \]

5. For \(6x^{-3/4}\):
\[ \frac{d}{dx}(6x^{-3/4}) = 6 \cdot -\frac{3}{4} x^{-7/4} = -\frac{9}{2} x^{-7/4}. \]

### Step 3: Combine the derivatives

Now, we combine all the derivatives:

\[ \frac{dy}{dx} = -\frac{1}{2x} + 5ax^{a-1} - 3a^x \ln(a) + \frac{2}{3} x^{-1/3} - \frac{9}{2} x^{-7/4}. \]

### Final Answer

Thus, the derivative of the given function with respect to \(x\) is:

\[ \frac{dy}{dx} = -\frac{1}{2x} + 5ax^{a-1} - 3a^x \ln(a) + \frac{2}{3} x^{-1/3} - \frac{9}{2} x^{-7/4}. \]