How to compute the derivative of a square root composite at a point

Key takeaway: A square root composition is a direct chain-rule job. Rewrite the root as a power first, then differentiate carefully.

Let

$u(x)=\sqrt{h(x)+3}=(h(x)+3)^{1/2}$

Step 1: Differentiate using the chain rule

$u'(x)=\frac12(h(x)+3)^{-1/2}\cdot h'(x)$

So

$u'(x)=\frac{h'(x)}{2\sqrt{h(x)+3}}$

Step 2: Evaluate at $x=1$

$u'(1)=\frac{h'(1)}{2\sqrt{h(1)+3}}$

Final answer

$\boxed{u'(1)=\frac{h'(1)}{2\sqrt{h(1)+3}}}$

You still need the values of $h(1)$ and $h'(1)$ to get a numerical result.

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Pitfalls the pros know 👇 Be careful not to differentiate $\sqrt{h(x)+3}$ as though it were just $\sqrt{h(x)}+\sqrt{3}$. The square root does not split across addition like that. Also, the denominator must stay under the root: $2\sqrt{h(x)+3}$.

What if the problem changes? If the inside expression gives $h(1)+3=0$, then the derivative is undefined at that point because the denominator becomes zero. If $h(1)+3<0$, the function is not real-valued there. So the domain matters before you differentiate at a specific point.

Tags: chain rule, square root function, derivative at a point