Question

Which of the following is derivative of \(\sqrt{a^{\sqrt{x}}}\) with respect to \(x\)?

• A. \(\dfrac{\sqrt{a^{\sqrt{x}}}}{2\sqrt{x}}\)
• B. \(\dfrac{\sqrt{a^{\sqrt{x}}}\log_e a}{2\sqrt{x}}\)
• C. \(\dfrac{a^{\sqrt{x}}\sqrt{a^{\sqrt{x}}}\log_e a}{2\sqrt{x}}\)
• D. \(\dfrac{\sqrt{a^{\sqrt{x}}}}{4\sqrt{x}\log_a e}\)

Hint:

First convert \(\sqrt{a^{\sqrt{x}}}\) into \(a^{\frac{\sqrt{x}}{2}}\), then apply the differentiation formula of \(a^{f(x)}\).

Answer 1

The Correct Option is D

Concept:
To differentiate an exponential function of the form \(a^{f(x)}\), we use: \[ \frac{d}{dx}\left(a^{f(x)}\right)=a^{f(x)}\log_e a \cdot f'(x) \]

Step 1: Write the given function. \[ y=\sqrt{a^{\sqrt{x}}} \] This can be written as: \[ y=\left(a^{\sqrt{x}}\right)^{\frac{1}{2}} \] \[ y=a^{\frac{\sqrt{x}}{2}} \]

Step 2: Differentiate using exponential rule. \[ \frac{dy}{dx}=a^{\frac{\sqrt{x}}{2}}\log_e a\cdot \frac{d}{dx}\left(\frac{\sqrt{x}}{2}\right) \]

Step 3: Differentiate the exponent. \[ \frac{d}{dx}\left(\frac{\sqrt{x}}{2}\right) =\frac{1}{2}\cdot \frac{1}{2\sqrt{x}} \] \[ =\frac{1}{4\sqrt{x}} \]

Step 4: Substitute. \[ \frac{dy}{dx}=a^{\frac{\sqrt{x}}{2}}\log_e a\cdot \frac{1}{4\sqrt{x}} \] \[ \frac{dy}{dx}=\frac{\sqrt{a^{\sqrt{x}}}\log_e a}{4\sqrt{x}} \] Now, \[ \log_a e=\frac{1}{\log_e a} \] Therefore, \[ \frac{\sqrt{a^{\sqrt{x}}}\log_e a}{4\sqrt{x}} = \frac{\sqrt{a^{\sqrt{x}}}}{4\sqrt{x}\log_a e} \] \[ \therefore \text{Correct Answer is (D)} \]