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

Options

A. `(8^(x))/(x^(8)) (log4 - (8)/(x))`
B. `(8^(x))/(x^(8)) (log8 - (8)/(x))`
C. `(8)/(x^(8)) (log8 - (8)/(x))`
D. `(8^(x))/(x^(8)) (log8 + (8)/(x))`

Explanation

To differentiate the function \( y = \frac{8^x}{x^8} \), we can follow these steps:

### Step 1: Rewrite the function
We start with the function:
\[ y = \frac{8^x}{x^8} \]

### Step 2: Take the natural logarithm of both sides
Taking the natural logarithm helps simplify the differentiation process:
\[ \ln y = \ln(8^x) - \ln(x^8) \]
Using the properties of logarithms, we can rewrite this as:
\[ \ln y = x \ln 8 - 8 \ln x \]

### Step 3: Differentiate both sides with respect to \( x \)
Now we differentiate both sides. Using implicit differentiation on the left side and the product rule on the right side:
\[ \frac{1}{y} \frac{dy}{dx} = \ln 8 - 8 \cdot \frac{1}{x} \]

### Step 4: Solve for \( \frac{dy}{dx} \)
To isolate \( \frac{dy}{dx} \), we multiply both sides by \( y \):
\[ \frac{dy}{dx} = y \left( \ln 8 - \frac{8}{x} \right) \]

### Step 5: Substitute back for \( y \)
Now we substitute back the value of \( y \):
\[ \frac{dy}{dx} = \frac{8^x}{x^8} \left( \ln 8 - \frac{8}{x} \right) \]

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

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