Q.Find $\frac{dy}{dx}$ in the following: $\sqrt{e^{\sqrt{x}}}, x > 0$

We are differentiating $y = \sqrt{e^{\sqrt{x}}}$. First, rewrite the square root as a power:

Now apply the chain rule. The outermost function is $e^{u}$ with $u = \frac{1}{2}\sqrt{x}$.

Step 1: Differentiate the outer exponential: $\frac{dy}{du} = e^{\frac{1}{2}\sqrt{x}}$.

Step 2: Differentiate $u = \frac{1}{2} x^{1/2}$: $\frac{du}{dx} = \frac{1}{2} \cdot \frac{1}{2} x^{-1/2} = \frac{1}{4\sqrt{x}}$. …

We have $y = \sqrt{e^{\sqrt{x}}}$. This is a composition of three functions: an outer square root, a middle exponential, and an inner square root. The Chain Rule says: differentiate from the outside in, multiplying by the derivative of each inner function.

Let’s rewrite the expression to make the nesting clear:

But it’s safer to keep the original form and apply the chain rule stepwise.

1. Outer function: square root

   Write $y = \sqrt{u}$ where $u = e^{\sqrt{x}}$.

   The derivative of $\sqrt{u}$ with respect to $u$ is $\frac{1}{2\sqrt{u}}$.

   So $\frac{dy}{du} = \frac{1}{2\sqrt{u}} = \frac{1}{2\sqrt{e^{\sqrt{x}}}}$.

2. Middle function: exponential

   Now $u = e^{v}$ where $v = \sqrt{x}$.

   The derivative of $e^{v}$ with respect to $v$ is $e^{v} = e^{\sqrt{x}}$.

   So $\frac{du}{dv} = e^{\sqrt{x}}$.

3. Inner function: square root

   Finally $v = \sqrt{x} = x^{1/2}$.

   Its derivative with respect to $x$ is $\frac{dv}{dx} = \frac{1}{2\sqrt{x}}$.

4. Multiply the chain

   By the Chain Rule:

Substitute each piece:

5. Simplify Combine the constants: $\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$. The exponential part: $\frac{e^{\sqrt{x}}}{\sqrt{e^{\sqrt{x}}}} = e^{\sqrt{x} - \frac{1}{2}\sqrt{x}} = e^{\frac{1}{2}\sqrt{x}}$. So we get: …

Method: Rewrite Nested Radicals/Exponentials as a Single Exponent Before Differentiating

Use this whenever a function stacks a radical on top of an exponential on top of another radical (or similar nested roots/powers) — rewriting in exponent form usually collapses several layers into one.

Steps

Step 1: Rewrite every radical as a fractional power

Combining the outer $\tfrac{1}{2}$ power with the exponential's exponent is the key simplification — it turns a three-layer nested function into a single exponential with a two-layer exponent.

Step 2: Differentiate the now-single exponential using the chain rule

For $y = e^{f(x)}$, $\dfrac{dy}{dx} = e^{f(x)}\cdot f'(x)$, where here $f(x) = \tfrac{1}{2}x^{1/2}$.

Step 3: Differentiate the exponent $f(x)$ itself using the power rule …

Common Mistakes

Mistake 1: Differentiating each nested layer separately instead of combining exponents first

Why it's wrong: it's not mathematically incorrect to chain through square-root, then exponential, then square-root again, but it triples the chances of an algebra slip compared to combining the exponents first. Correct approach: always rewrite nested radicals/exponentials as a single power/exponent before differentiating, when possible.

Mistake 2: Forgetting the extra $\tfrac12$ factor from the outer square root …