Math · Calculus I · Worked example

Differentiate e^(−x²) with the chain rule

Differentiate y = e^(−x²), then use the derivative to find where the curve rises and falls.

y = e^{- x^{2}}

Name the layers

Work out e^(−x²) for x = 2: first −x² = −(2²) = −4, then e^(−4). The exponent is worked out first, so it is the inside, and raising e to that power is the outside. Note that −x² means −(x²), so it is never positive.

u = - x^{2}, y = e^{u}

Differentiate the outside

eᵘ is its own derivative, so dy/du = eᵘ. The exponent is copied unchanged.

\frac{d y}{d u} = e^{u}

Differentiate the inside

The derivative of −x² is −2x.

\frac{d u}{d x} = - 2 x

Multiply, then put the inside back

Multiply the rates, replace u with −x², and write the factor −2x in front.

\frac{d y}{d x} = e^{- x^{2}} \cdot (- 2 x) = - 2 x e^{- x^{2}}

Read the sign of the derivative

e^(−x²) is positive for every x, so the derivative has the same sign as −2x. For x < 0 the derivative is positive and the curve rises. At x = 0 it is 0, the flat top of the bell at y = 1. For x > 0 it is negative and the curve falls.

Result

dy/dx = −2x e^(−x²). The curve rises for x < 0, levels off at its peak (0, 1) and falls for x > 0.

\frac{d y}{d x} = - 2 x e^{- x^{2}}

Your turn

Differentiate y = ln(x² + 1).

Show the answer and explanation

dy/dx = 2x/(x² + 1).

The inside is u = x² + 1 and the outside is ln u, whose derivative is 1/u. Multiply by the inside’s derivative, 2x: (1/(x² + 1))·2x = 2x/(x² + 1). Since x² + 1 is positive for every x, the logarithm and its derivative are defined everywhere.

\begin{array}{l} y = ln (x^{2} + 1) \\ \frac{d y}{d x} = \frac{1}{x^{2} + 1} \cdot 2 x \\ \frac{d y}{d x} = \frac{2 x}{x^{2} + 1} \end{array}

Keep exploring

Open the curve and its derivative in Graph: g(x) = −2xe^(−x²) is positive left of 0, crosses zero at the peak x = 0 and is negative to the right.

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Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

OpenStax Calculus Volume 1 — The chain rule

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Check the derivative in Math See the curve and its derivative in Graph Open worked example on a board Chain rule in Math Reference

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