Question

If $y = \sqrt{e^{\sqrt{x}}}$, then $\frac{dy}{dx} =$

• A. $\frac{\sqrt{e^{\sqrt{x}}}}{4\sqrt{x}}$
• B. $\frac{e^{\sqrt{x}}}{4\sqrt{x}}$
• C. $\frac{\sqrt{e^{\sqrt{x}}}}{2\sqrt{x}}$
• D. $\frac{e^{\sqrt{x}}}{2\sqrt{x}}$

Hint:

Algebraic simplification before calculus is a powerful tool! Rewriting nested radicals as fractional exponents usually makes the chain rule much less messy and prevents silly mistakes.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We must find the first derivative of a composite exponential function with respect to $x$ using the chain rule.

Step 2: Key Formula or Approach:
Simplify the expression using exponent rules before differentiating. Remember that $\sqrt{a} = a^{1/2}$. Then, apply the chain rule: $\frac{d}{dx} [e^{u}] = e^{u} \cdot \frac{du}{dx}$.

Step 3: Detailed Explanation:
The given function is:
$$y = \sqrt{e^{\sqrt{x}}}$$ Rewrite the square root as a fractional exponent:
$$y = (e^{\sqrt{x}})^{1/2}$$ Using the power of a power property $(a^m)^n = a^{mn}$:
$$y = e^{\frac{1}{2}\sqrt{x}}$$ Now, differentiate with respect to $x$ using the chain rule:
$$\frac{dy}{dx} = e^{\frac{1}{2}\sqrt{x}} \cdot \frac{d}{dx} \left( \frac{1}{2}\sqrt{x} \right)$$ The derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$:
$$\frac{dy}{dx} = e^{\frac{1}{2}\sqrt{x}} \cdot \frac{1}{2} \left( \frac{1}{2\sqrt{x}} \right)$$ $$\frac{dy}{dx} = e^{\frac{1}{2}\sqrt{x}} \cdot \frac{1}{4\sqrt{x}}$$ Convert the exponential term back to its original radical form:
$$\frac{dy}{dx} = \sqrt{e^{\sqrt{x}}} \cdot \frac{1}{4\sqrt{x}} = \frac{\sqrt{e^{\sqrt{x}}}}{4\sqrt{x}}$$

Step 4: Final Answer:
The derivative is $\frac{\sqrt{e^{\sqrt{x}}}}{4\sqrt{x}}$, matching option (A).