Derivative of e^(2x)

Answer

2e^{2x}

Derivative of e^(2x)

Rule Used: Chain Rule
Steps: Identify a composite function in $e^{2 \cdot x}$ : use the chain rule $\frac{d}{dx} f(g(x))$ = f'(g(x))·g'(x). The inner derivative is required because a stretch or shift of the input changes how fast the output moves.
Differentiate the outer function, then multiply by the inner derivative.
Final answer: $2 \cdot e^{2 \cdot x}$

d/dx e^(2x) = 2*e^(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 e^(2x)

Derivative ExponentialIntermediate

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Derivative of e^(2x) = 2*e^(2*x) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-e-to-the-2x

Derivative

e^(2x) → 2*e^(2*x)engine: 2*e^(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

e^(2x) grows twice as fast in the exponent as e^x. The chain rule multiplies e^(2x) by the derivative of 2x, which is 2. The 2 stays in front and does not move into the exponent. At x = 0 the function equals 1 and the slope equals 2, which is the numerical check that the factor survived.

Apply the chain rule to e^(2x). The form to remember, already simplified, is 2*e^(2*x).

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

At x = 0, substitute into the simplified form 2*e^(2*x). That number is the slope of e^(2x) there; a secant over a small step in x lands next to it.

Worked derivative

Write f(x) = e^(2x).
Apply the chain rule to e^(2x). The form to remember, already simplified, is 2*e^(2*x).
At x = 0, substitute into the simplified form 2*e^(2*x). That number is the slope of e^(2x) there; a secant over a small step in x lands next to it.

Common mistakes

Answering e^(2x) instead of 2e^(2x)

The answer e^(2x) is missing the inner derivative 2. The exponent is not x, so the factor 2 has to appear in the slope.

Treating e^(2x) as if only the base mattered

Only e^x is its own derivative with no extra factor. A coefficient in the exponent, or a base other than e, changes the answer.

Chain rule — The rule page lists this example and the others in the same family.
Exponential derivatives — The rule page lists this example and the others in the same family.
Derivative of e^x — Same family of differentiation, different function.
Derivative of e^(-x) — Same family of differentiation, different function.
Derivative of 2^x — Same family of differentiation, different function.
Derivative of 10^x — Same family of differentiation, different function.

Frequently asked questions

What is the derivative of e^(2x)?

2*e^(2*x). e^(2x) grows twice as fast in the exponent as e^x. The chain rule multiplies e^(2x) by the derivative of 2x, which is 2. The 2 stays in front and does not move into the exponent. At x = 0 the function equals 1 and the slope equals 2, which is the numerical check that the factor survived.