Simplifying the Function

First, simplify the expression $ \sqrt{a^{\sqrt{x}}} $ using exponent rules.

• Let $ f(x) = \sqrt{a^{\sqrt{x}}} $.
• Rewrite using exponents: $ f(x) = \left(a^{x^{1/2}}\right)^{1/2} = a^{\frac{1}{2}x^{1/2}} $.

Derivative Calculation

To find the derivative $ \frac{df}{dx} $, apply the chain rule and the derivative rule for exponential functions $ \frac{d}{dx}(a^u) = a^u \ln(a) \frac{du}{dx} $.

• In $ f(x) = a^{\frac{1}{2}x^{1/2}} $, let the exponent be $ u = \frac{1}{2}x^{1/2} $.
• Find the derivative of $ u $ with respect to $ x $:

  $ \frac{du}{dx} = \frac{d}{dx}\left(\frac{1}{2}x^{1/2}\right) $. Using the power rule $ \frac{d}{dx}(x^n) = nx^{n-1} $, we get:

  $ \frac{du}{dx} = \frac{1}{2} \cdot \frac{1}{2}x^{1/2 - 1} = \frac{1}{4}x^{-1/2} = \frac{1}{4\sqrt{x}} $.

• Now apply the derivative rule for $ a^u $:

  $ \frac{df}{dx} = a^u \ln(a) \frac{du}{dx} $.

• Substitute $ u = \frac{1}{2}\sqrt{x} $ and $ \frac{du}{dx} = \frac{1}{4\sqrt{x}} $:

  $ \frac{df}{dx} = a^{\frac{1}{2}\sqrt{x}} \cdot \ln(a) \cdot \frac{1}{4\sqrt{x}} $.

• Rewrite $ a^{\frac{1}{2}\sqrt{x}} $ back to the original form $ \sqrt{a^{\sqrt{x}}} $:

  $ \frac{df}{dx} = \sqrt{a^{\sqrt{x}}} \cdot \ln(a) \cdot \frac{1}{4\sqrt{x}} = \frac{\sqrt{a^{\sqrt{x}}} \ln a}{4\sqrt{x}} $.

Comparing Result with Options

The calculated derivative is $ \frac{\sqrt{a^{\sqrt{x}}} \ln a}{4\sqrt{x}} $. Let's examine the options.

• Option 4 is $ \frac{\sqrt{a^{\sqrt{x}}}}{4\sqrt{x} \log_a e} $.
• Use the change of base formula for logarithms: $ \log_a e = \frac{\ln e}{\ln a} = \frac{1}{\ln a} $.
• This implies $ \frac{1}{\log_a e} = \ln a $.
• Substitute this into Option 4:

  $ \frac{\sqrt{a^{\sqrt{x}}}}{4\sqrt{x} \log_a e} = \frac{\sqrt{a^{\sqrt{x}}}}{4\sqrt{x}} \cdot \frac{1}{\log_a e} = \frac{\sqrt{a^{\sqrt{x}}}}{4\sqrt{x}} \cdot \ln a = \frac{\sqrt{a^{\sqrt{x}}} \ln a}{4\sqrt{x}} $.

• This result exactly matches our calculated derivative.