How to differentiate a function defined by a square root composition

Expert intro: Square roots inside a function are a classic chain rule setup. The key is to treat the inside as the inner variable and then evaluate at the point carefully.

Detailed walkthrough

We have

$u(x)=h(\sqrt{x})$

and need $u'(4)$.

Step 1: Differentiate using the chain rule

Let the inside be $\sqrt{x}$. Then

$u'(x)=h'(\sqrt{x})\cdot \frac{d}{dx}(\sqrt{x})$

Since

$\frac{d}{dx}(\sqrt{x})=\frac{1}{2\sqrt{x}}$

we get

$u'(x)=h'(\sqrt{x})\cdot \frac{1}{2\sqrt{x}}$

Step 2: Evaluate at $x=4$

$u'(4)=h'(\sqrt{4})\cdot \frac{1}{2\sqrt{4}}$

$u'(4)=h'(2)\cdot \frac{1}{4}$

Final answer

$\boxed{u'(4)=\frac{1}{4}h'(2)}$

💡 Pitfall guide

A frequent mistake is to write $h'(x)$ instead of $h'(\sqrt{x})$. The derivative of the outer function is evaluated at the inner input, not at $x$ directly. Another slip is forgetting the derivative of $\sqrt{x}$ itself.

🔄 Real-world variant

If the function were $u(x)=h(ax)$ for a constant $a$, then the derivative would be

$u'(x)=ah'(ax)$

So with a square root, the extra factor is $\frac{1}{2\sqrt{x}}$ instead of a constant.

🔍 Related terms

chain rule, square root derivative, composition