Derivative of x^2+1

Answer

2 \cdot x

Derivative of x^2+1

Rule Used: Sum Rule
Steps: Differentiate $x^2$ +1 term by term. The sum rule says the derivative of a sum is the sum of the derivatives, and the derivative of a constant term is 0, so that term disappears.
Final answer: $2 \cdot x$

d/dx x^2+1 = 2*x. The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.

Instant step-by-step solution for x^2+1

Derivative PolynomialIntermediate

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Derivative of x^2+1 = 2*x Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-x-squared-plus-1

Derivative

x^2+1 → 2*xengine: 2*x

Graph of the function and its derivative

The solid curve is f. The dashed curve is f′. Vertical gaps are asymptotes or jumps, not extra zeros.

Why the derivative has this form

x^2 + 1 sits one unit above x^2. The added 1 changes height and does not change slope, so the derivative is again 2x. It matches the derivative of x^2 − 1 for the same reason, and it does not match the function x^2 + 1 itself. A quick check: at x = 0 both shifted parabolas have a horizontal tangent, and 2x is 0 there.

Apply the power rule to x^2+1. The form to remember, already simplified, is 2*x.

The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.

At x = 3, substitute into the simplified form 2*x. That number is the slope of x^2+1 there; a secant over a small step in x lands next to it.

Worked derivative

Write f(x) = x^2+1.
Apply the power rule to x^2+1. The form to remember, already simplified, is 2*x.
At x = 3, substitute into the simplified form 2*x. That number is the slope of x^2+1 there; a secant over a small step in x lands next to it.

Common mistakes

Answering 2*x+1

2x + 1 keeps the constant alive. Constants disappear when you differentiate; they do not turn into 1.

Differentiating the constant in x^2+1

The constant in x^2+1 changes height only. Its derivative is 0, so it cannot leave a +1 or a −1 in the slope.

Power rule — The rule page lists this example and the others in the same family.
Derivative of x^2-1 — Same family of differentiation, different function.

Frequently asked questions

What is the derivative of x^2+1?

2*x. x^2 + 1 sits one unit above x^2. The added 1 changes height and does not change slope, so the derivative is again 2x. It matches the derivative of x^2 − 1 for the same reason, and it does not match the function x^2 + 1 itself. A quick check: at x = 0 both shifted parabolas have a horizontal tangent, and 2x is 0 there.