Differentiate the following w.r.t. x :
`(1/2)^(x)`

Answer

To differentiate the function \( y = \left( \frac{1}{2} \right)^x \) with respect to \( x \), we can follow these steps:

### Step 1: Rewrite the function using properties of exponents
We can express \( \left( \frac{1}{2} \right)^x \) in terms of the natural exponential function:
\[ y = \left( \frac{1}{2} \right)^x = 2^{-x} \]

### Step 2: Differentiate using the chain rule
Now we can differentiate \( y = 2^{-x} \) using the chain rule. The derivative of \( a^x \) is given by:
\[ \frac{d}{dx}(a^x) = a^x \ln(a) \]
So, for \( y = 2^{-x} \):
\[ \frac{dy}{dx} = \frac{d}{dx}(2^{-x}) = 2^{-x} \cdot \ln(2^{-1}) \]

### Step 3: Simplify the logarithm
Using the property of logarithms \( \ln(a^{-1}) = -\ln(a) \):
\[ \ln(2^{-1}) = -\ln(2) \]
Thus, we can substitute this back into our derivative:
\[ \frac{dy}{dx} = 2^{-x} \cdot (-\ln(2)) = -\ln(2) \cdot 2^{-x} \]

### Final Answer
The derivative of \( y = \left( \frac{1}{2} \right)^x \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -\frac{\ln(2)}{2^x} \]

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