If `y = log _(5) (log _(7) x ),` then find `(dy )/(dx)`

Options

A. `(1)/(x log 5. log x)`
B. `(-1)/(x log .5 log x)`
C. `(1)/(x log x)`
D. `(1)/(x log 7. log x)`

Answer

To find the derivative of the function \( y = \log_5(\log_7(x)) \), we will apply the chain rule and the properties of logarithms. Here’s a step-by-step solution:

### Step 1: Rewrite the logarithm in terms of natural logarithms
We can use the change of base formula for logarithms:
\[ \log_a(b) = \frac{\log_e(b)}{\log_e(a)} \]
Thus, we can rewrite \( y \) as:
\[ y = \frac{\log_e(\log_7(x))}{\log_e(5)} \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we apply the chain rule. The derivative of \( y \) with respect to \( x \) is given by:
\[ \frac{dy}{dx} = \frac{1}{\log_e(5)} \cdot \frac{d}{dx}[\log_e(\log_7(x))] \]

### Step 3: Differentiate \( \log_e(\log_7(x)) \)
Using the chain rule again, we have:
\[ \frac{d}{dx}[\log_e(\log_7(x))] = \frac{1}{\log_7(x)} \cdot \frac{d}{dx}[\log_7(x)] \]

### Step 4: Differentiate \( \log_7(x) \)
Using the change of base formula again:
\[ \log_7(x) = \frac{\log_e(x)}{\log_e(7)} \]
Thus, its derivative is:
\[ \frac{d}{dx}[\log_7(x)] = \frac{1}{x \cdot \log_e(7)} \]

### Step 5: Substitute back into the derivative
Now substitute this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{\log_e(5)} \cdot \frac{1}{\log_7(x)} \cdot \frac{1}{x \cdot \log_e(7)} \]

### Step 6: Simplify the expression
Now we can simplify:
\[ \frac{dy}{dx} = \frac{1}{x \cdot \log_e(5) \cdot \log_7(x) \cdot \log_e(7)} \]

### Final Expression
Thus, the final expression for the derivative is:
\[ \frac{dy}{dx} = \frac{1}{x \cdot \log_e(5) \cdot \log_7(x) \cdot \log_e(7)} \]