Derivative of 2^x

Answer

2^{x}\ln(2)

Derivative of 2^x

Rule Used: Exponential Rule
Steps: Recognize the exponential 2^x. For e^x the derivative equals the original function. For a^x the factor $\ln(a)$ appears, because a^x = $e^{x\cdot \ln(a)}$ .
Apply the rule and keep that constant factor.
Final answer: 2^x* $\ln(2)$

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

Instant step-by-step solution for 2^x

Derivative ExponentialIntermediate

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

Derivative

2^x → 2^x*ln(2)engine: 2^x*ln(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

2^x is not e^x. Write it as e^(x ln 2); the chain rule then multiplies by ln 2. The derivative is 2^x ln 2, written 2^x*ln(2). The ln 2 is a constant about 0.693, which is why 2^x grows more slowly than e^x.

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

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^x*ln(2). That number is the slope of 2^x there; a secant over a small step in x lands next to it.

Worked derivative

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

Common mistakes

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

2^x without ln 2 treats every exponential as its own derivative. Only base e has that property.

Treating 2^x 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.

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 e^(2x) — Same family of differentiation, different function.
Derivative of 10^x — Same family of differentiation, different function.

Frequently asked questions

What is the derivative of 2^x?

2^x*ln(2). 2^x is not e^x. Write it as e^(x ln 2); the chain rule then multiplies by ln 2. The derivative is 2^x ln 2, written 2^x*ln(2). The ln 2 is a constant about 0.693, which is why 2^x grows more slowly than e^x.