Derivative of 2x is 2

Answer

2

Derivative of 2x

Rule Used: Constant Multiple Rule
Steps: $2 \cdot x$ is a constant times a function of x. The constant is a fixed stretch of the slope, so it factors out and the remaining function is differentiated.
Final answer: 2

The derivative of 2x is 2 for every real x.

Instant step-by-step solution for 2x

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Derivative

2x → 2engine: 2

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

The graph of y = 2x is a straight line through the origin with slope 2. A derivative is a slope, so the answer cannot depend on x: every tangent line to a straight line is the line itself. Any expression that still contains x, such as 2x or x^2, describes a curve whose steepness changes from point to point, and 2x is not a curve.

The difference quotient makes this exact. f(x+h) - f(x) = 2(x+h) - 2x = 2h, and dividing by h leaves 2 before the limit is even taken. The h cancels completely, so there is nothing left for h -> 0 to change. Replacing the coefficient 2 by any constant k gives k, which is the constant multiple rule in its simplest case.

Adding a constant does not change the slope either, so the derivative of 2x + 7 is still 2. Multiplying by a function of x is different: the derivative of 2x sin(x) needs the product rule and equals 2 sin(x) + 2x cos(x), which varies with x. The 2 in 2x is a coefficient of x, not a second function composed with it.

Difference quotient, checked two other ways

Set f(x) = 2x and form the difference quotient (f(x+h) - f(x))/h.
Expand f(x+h) = 2(x+h) = 2x + 2h.
Subtract f(x): (2x + 2h) - 2x = 2h.
Divide by h, with h not zero: 2h/h = 2. The quotient is already constant.
Take the limit as h approaches 0. The limit of the constant 2 is 2.
Check with the power rule: 2x = 2x^1, so the derivative is 2·1·x^0 = 2.
Check with the constant multiple rule: d/dx [2·x] = 2·d/dx[x] = 2·1 = 2.

Worked example

Tangent line to y = 2x at x = 3, and to the shifted line y = 2x + 7

At x = 3 the point on y = 2x is (3, 6), and the derivative gives a slope of 2.
The tangent line is y - 6 = 2(x - 3), which simplifies back to y = 2x. A line is its own tangent.
For y = 2x + 7 the point is (3, 13) and the slope is still 2, giving y - 13 = 2(x - 3), or y = 2x + 7.
A secant from x = 3 to x = 3.001 rises by 0.002 over 0.001, a slope of exactly 2. The ratio is 2 at every starting point.

Answer: 2

Common mistakes

Answering 2x

2x is the function, not its rate of change. The difference quotient cancels x completely, so a correct answer here has no x in it. If a computed derivative of a linear function still contains the variable, the subtraction step was skipped.

Answering 1 or x

The derivative of x is 1, and the constant multiple rule then multiplies that 1 by 2. Dropping the coefficient gives 1; keeping x gives a slope that grows with x. Both contradict the straight-line graph.

Inventing a chain-rule factor

2x is a product of a constant and x, not a composition. There is no inner function whose derivative must be brought down. Chain-rule factors appear for expressions like sin(2x), where the inner function 2x genuinely contributes a factor of 2.

Derivative of x^2 — 2x. A slope that really does depend on x, unlike this page.
Power rule — The rule page lists this example and the others in the same family.
Derivative of 5x — Same family of differentiation, different function.
Derivative of x+1 — Same family of differentiation, different function.
Derivative of 1-x — Same family of differentiation, different function.
Derivative of -x — Same family of differentiation, different function.