The differential coefficient of `a^(log10" cosec"^(-1)x)`, is

Options

A. `(a^(log10("cosec"^(-1)x)))/("cosec"^(-1)x)(1)/(xsqrt(x^(2)-1))log_(10)a`
B. `-(a^(log10("cosec"^(-1)x)))/("cosec"^(-1)x).(1)/(|x|sqrt(x^(2)-1))log_(10)a`
C. `(-a^(log10("cosec"^(-1)x)))/("cosec"^(-1)x).(1)/(|x|sqrt(x^(2)-1))log_(a)10`
D. `(a^(log10"cosec"^(-1)x))/("cosec"^(-1)x).(1)/(xsqrt(x^(2)-1))log_(a)10`

Answer

To find the differential coefficient of \( y = a^{\log_{10}(\csc^{-1}(x))} \), we will follow these steps:

### Step 1: Rewrite the function
Let \( y = a^{\log_{10}(\csc^{-1}(x))} \).

### Step 2: Differentiate using the chain rule
Using the property of differentiation, we know that:
\[ \frac{dy}{dx} = a^{\log_{10}(\csc^{-1}(x))} \cdot \log(a) \cdot \frac{d}{dx}[\log_{10}(\csc^{-1}(x))] \]

### Step 3: Differentiate \( \log_{10}(\csc^{-1}(x)) \)
Using the change of base formula:
\[ \frac{d}{dx}[\log_{10}(\csc^{-1}(x))] = \frac{1}{\csc^{-1}(x) \ln(10)} \cdot \frac{d}{dx}[\csc^{-1}(x)] \]

### Step 4: Differentiate \( \csc^{-1}(x) \)
The derivative of \( \csc^{-1}(x) \) is:
\[ \frac{d}{dx}[\csc^{-1}(x)] = -\frac{1}{|x| \sqrt{x^2 - 1}} \]

### Step 5: Substitute back into the derivative
Now substituting this back into our expression:
\[ \frac{d}{dx}[\log_{10}(\csc^{-1}(x))] = \frac{1}{\csc^{-1}(x) \ln(10)} \cdot \left(-\frac{1}{|x| \sqrt{x^2 - 1}}\right) \]

### Step 6: Combine the results
Now substituting this into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = a^{\log_{10}(\csc^{-1}(x))} \cdot \log(a) \cdot \left(-\frac{1}{\csc^{-1}(x) \ln(10)} \cdot \frac{1}{|x| \sqrt{x^2 - 1}}\right) \]

### Step 7: Final expression
Thus, we can write:
\[ \frac{dy}{dx} = -\frac{a^{\log_{10}(\csc^{-1}(x))} \cdot \log(a)}{\csc^{-1}(x) \cdot |x| \cdot \sqrt{x^2 - 1} \cdot \ln(10)} \]

This is the required differential coefficient.