What is the derivative of `log_(x)5` respect to `log_(5)x` ?

Options

A. `-(log_(5)x)^(-2)`
B. `(log_(5)x)^(-2)`
C. `-(log_(x)5)^(-2)`
D. `(log_(5)5)^(-2)`

Answer

To find the derivative of \( \log_{x} 5 \) with respect to \( \log_{5} x \), we can follow these steps:

### Step 1: Rewrite the logarithmic expressions
We can express \( \log_{x} 5 \) and \( \log_{5} x \) in terms of natural logarithms (base \( e \)):
\[ u = \log_{x} 5 = \frac{\log_{e} 5}{\log_{e} x} \]
\[ v = \log_{5} x = \frac{\log_{e} x}{\log_{e} 5} \]

### Step 2: Differentiate \( u \) with respect to \( x \)
Using the quotient rule, we differentiate \( u \):
\[ \frac{du}{dx} = \frac{d}{dx} \left( \frac{\log_{e} 5}{\log_{e} x} \right) = \log_{e} 5 \cdot \frac{d}{dx} \left( \frac{1}{\log_{e} x} \right) \]
The derivative of \( \frac{1}{\log_{e} x} \) using the chain rule is:
\[ \frac{d}{dx} \left( \frac{1}{\log_{e} x} \right) = -\frac{1}{(\log_{e} x)^2} \cdot \frac{1}{x} \]
Thus,
\[ \frac{du}{dx} = -\frac{\log_{e} 5}{x (\log_{e} x)^2} \]

### Step 3: Differentiate \( v \) with respect to \( x \)
Now we differentiate \( v \):
\[ \frac{dv}{dx} = \frac{d}{dx} \left( \frac{\log_{e} x}{\log_{e} 5} \right) = \frac{1}{\log_{e} 5} \cdot \frac{1}{x} \]

### Step 4: Find \( \frac{du}{dv} \)
Now we can find the derivative \( \frac{du}{dv} \):
\[ \frac{du}{dv} = \frac{\frac{du}{dx}}{\frac{dv}{dx}} = \frac{-\frac{\log_{e} 5}{x (\log_{e} x)^2}}{\frac{1}{x \log_{e} 5}} \]
This simplifies to:
\[ \frac{du}{dv} = -\frac{\log_{e} 5}{(\log_{e} x)^2} \cdot \frac{x \log_{e} 5}{x} = -\frac{(\log_{e} 5)^2}{(\log_{e} x)^2} \]

### Step 5: Final expression
Thus, the derivative of \( \log_{x} 5 \) with respect to \( \log_{5} x \) is:
\[ \frac{du}{dv} = -\frac{(\log_{e} 5)^2}{(\log_{e} x)^2} \]