If `y = 36^(log_(6)x)`, then `(dy)/(dx)` is

Options

A. 2x
B. 6x
C. 36x
D. log 6x

Answer

To find the derivative \( \frac{dy}{dx} \) of the function \( y = 36^{\log_6 x} \), we can follow these steps:

### Step 1: Rewrite the base
We can express 36 in terms of base 6:
\[ 36 = 6^2 \]
Thus, we can rewrite \( y \):
\[ y = (6^2)^{\log_6 x} \]

### Step 2: Apply the power of a power property
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \), we can simplify \( y \):
\[ y = 6^{2 \cdot \log_6 x} \]

### Step 3: Use the logarithmic identity
Using the identity \( a^{\log_a b} = b \), we can simplify further:
\[ y = x^2 \]

### Step 4: Differentiate \( y \)
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x^2) = 2x \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 2x \]

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