Find the derivative of \( y=3^{\sqrt{x}} \) with respect to \( x \). \( \frac{d y}{d x}=\square \)

We start with the function \[ y = 3^{\sqrt{x}} \] recognizing that it is of the form \( y = a^{u(x)} \) where \( a = 3 \) and \( u(x) = \sqrt{x} \). **Step 1:** Differentiate using the chain rule for exponential functions. The derivative of \( a^{u(x)} \) is \[ \frac{dy}{dx} = a^{u(x)} \ln(a) \cdot u'(x) \] Substitute \( a = 3 \) and \( u(x) = \sqrt{x} \): \[ \frac{dy}{dx} = 3^{\sqrt{x}} \ln(3) \cdot \frac{d}{dx} \left( \sqrt{x} \right) \] **Step 2:** Compute the derivative of \( \sqrt{x} \). Recall that \[ \sqrt{x} = x^{\frac{1}{2}} \] Using the power rule: \[ \frac{d}{dx} \left( x^{\frac{1}{2}} \right) = \frac{1}{2} x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}} \] **Step 3:** Substitute the derivative of \( \sqrt{x} \) back into the expression. \[ \frac{dy}{dx} = 3^{\sqrt{x}} \ln(3) \cdot \frac{1}{2\sqrt{x}} \] Thus, the derivative is \[ \frac{dy}{dx} = \frac{3^{\sqrt{x}} \ln(3)}{2\sqrt{x}} \]