Derivative of `sqrt(x^2+1)`

Step-by-step differentiation

Answer

\frac{d}{d x} [x^{2} + 1]

\frac{x}{x ^{2} + 1}

Step-by-step solution

1Chain Rule — sqrt
$\frac{d}{d x} [x^{2} + 1] = \frac{1}{2 x ^{2} + 1} \cdot \frac{d}{d x} [x^{2} + 1]$
The derivative of $u$ is $\frac{1}{2 u}$ . Here $u = x^{2} + 1$ . Multiply by $u^{'}$ (chain rule).
↳Find u′: Differentiate the inner function, $u = x^{2} + 1$ :
2Sum Rule
$\frac{d}{d x} [x^{2} + 1] = \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$
Use the sum rule: differentiate each term on its own, then add the results.
3Power Rule
$\frac{d}{d x} [x^{2}] = 2 \cdot x^{2 - 1}$
The exponent is $2$ . Bring it down in front as a coefficient, then reduce the power by $1$ .
4Constant Rule
$\frac{d}{d x} [1] = 0$
This term doesn't involve $x$ at all — it's constant, so its derivative is $0$ .
=Combine
$\frac{d}{d x} [x^{2} + 1] = 2 x + 0 = 2 x$
Both terms are differentiated. Add them: $2 x + 0 = 2 x$ .
=Combine
$\frac{d}{d x} [x^{2} + 1] = \frac{\frac{1}{2}}{x ^{2} + 1} \cdot 2 x = \frac{x}{x ^{2} + 1}$
With $u^{'} = 2 x$ , multiply by the outer derivative to complete the chain rule.
✓Final Answer
$\frac{d}{d x} [x^{2} + 1] = \frac{x}{x ^{2} + 1}$
The first derivative simplifies to $\frac{x}{x ^{2} + 1}$ .

Understanding this derivative

Differentiating sqrt(x²+1) exercises 4 distinct techniques — the chain rule, the power rule, the square root rule and the sum and difference rules. Problems like this one are useful practice precisely because the rules have to be combined in the right order rather than applied in isolation.

The chain rule handles the composite structure here: differentiate the outer function while leaving the inner expression alone, then multiply by the derivative of that inner expression. Forgetting that final multiplication is one of the most common mistakes in differentiation. The power rule is the workhorse step: bring the exponent down as a coefficient and reduce the exponent by one. A square root is a power of one-half in disguise, so the power rule gives d/dx[√x] = 1/(2√x); the chain rule extends this to roots of larger expressions. Sums and differences differentiate term by term, so the expression splits into independent pieces that are each handled with their own rule.

A good way to verify a derivative by hand is to compare it against the step-by-step breakdown above — each step names the rule being applied, so you can pinpoint exactly where your own work diverges if the answers differ.

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Derivative of sqrt(x^2+1)

Step-by-step solution

Understanding this derivative

Derivative of `sqrt(x^2+1)`