Differentiate `(8^(x))/(x^(8))`

Explanation

To differentiate the function \( y = \frac{8^x}{x^8} \), we can use logarithmic differentiation. Here is a step-by-step solution:

### Step 1: Define the function
Let
\[ y = \frac{8^x}{x^8} \]

### Step 2: Take the natural logarithm of both sides
Taking the logarithm of both sides gives:
\[ \log y = \log \left( \frac{8^x}{x^8} \right) \]

### Step 3: Apply logarithmic properties
Using the properties of logarithms, we can rewrite the right-hand side:
\[ \log y = \log(8^x) - \log(x^8) \]
This simplifies to:
\[ \log y = x \log 8 - 8 \log x \]

### Step 4: Differentiate both sides with respect to \( x \)
Now, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(\log y) = \frac{d}{dx}(x \log 8 - 8 \log x) \]
Using the chain rule on the left side, we get:
\[ \frac{1}{y} \frac{dy}{dx} = \log 8 - 8 \cdot \frac{1}{x} \]

### Step 5: Solve for \( \frac{dy}{dx} \)
Now, we can solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = y \left( \log 8 - \frac{8}{x} \right) \]

### Step 6: Substitute back for \( y \)
Substituting back the expression for \( y \):
\[ \frac{dy}{dx} = \frac{8^x}{x^8} \left( \log 8 - \frac{8}{x} \right) \]

### Final Answer
Thus, the derivative of \( y = \frac{8^x}{x^8} \) is:
\[ \frac{dy}{dx} = \frac{8^x}{x^8} \left( \log 8 - \frac{8}{x} \right) \]

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