AP Calculus AB and BC

Derivative of e^(x^2): Answer, Chain Rule, Mistakes

The derivative of e to the x squared with respect to x is 2x times e to the x squared. The exponential differentiates to itself and the chain rule multiplies by the derivative of the exponent, which is 2x. Note this is a different function from e to the 2x.

ddx[ex2]=2xex2\frac{d}{dx}\left[e^{x^{2}}\right] = 2xe^{x^{2}}

How to differentiate e^(x^2)

The inner function is the exponent x2x^{2}, whose derivative is 2x2x, and the outer function is the exponential.

ddxex2=ex2⋅2x=2xex2\frac{d}{dx}e^{x^{2}} = e^{x^{2}}\cdot 2x = 2xe^{x^{2}}

Not the same as e^(2x)

ex2e^{x^{2}} has the square in the EXPONENT and differentiates to 2xex22xe^{x^{2}}. e2xe^{2x} has a linear exponent and differentiates to 2e2x2e^{2x}. The notation differs by one position and the answers are different functions.

Why it has no elementary antiderivative

Differentiating this function is routine, but antidifferentiating it is impossible in elementary terms. There is no combination of the standard functions whose derivative is ex2e^{x^{2}}.

Its close relative e−x2e^{-x^{2}} is the shape of the normal distribution, which is why statistics uses numerical tables rather than a formula. On BC, such integrands are handled with a Taylor series instead.

Where the derivative of e^(x^2) shows up on the AP exam

Topic 3.1 on both exams. It also appears as a Fundamental Theorem question: if F(x)=∫0xet2 dtF(x) = \int_{0}^{x} e^{t^{2}}\,dt then F′(x)=ex2F'(x) = e^{x^{2}}, which is answerable even though the integral itself is not.

Common mistakes with the derivative of e^(x^2)

  • Answering ex2e^{x^{2}} and forgetting the chain rule factor.
  • Answering 2xex2−12xe^{x^{2}-1}, applying the power rule to an exponential.
  • Confusing it with (ex)2=e2x\left(e^{x}\right)^{2} = e^{2x}, whose derivative is 2e2x2e^{2x}.

Check yourself, not just the answer

Type derivatives and get graded on mathematical equivalence, with rule-level hints when you miss, in the Derivative Practice Checker.

Frequently asked questions

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

It is 2xex22xe^{x^{2}}.

How is e^(x^2) different from e^(2x)?

The exponent is squared rather than doubled. Their derivatives are 2xex22xe^{x^{2}} and 2e2x2e^{2x}, which are different functions.

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

There is no elementary antiderivative. On BC you would represent it as a Taylor series or evaluate it numerically.

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

It is −2xe−x2-2xe^{-x^{2}}; the chain rule brings down −2x-2x from the exponent.