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Derivative of x2ex\frac{x^{2}}{e^{x}}

The derivative of x2ex\frac{x^{2}}{e^{x}} is \frac{2 x e^{x} - x^{2} e^{x}}{e^{x}^{2}}.

f(x) =
\frac{d}{dx}\left[\frac{x^{2}}{e^{x}}\right] = \frac{2 x e^{x} - x^{2} e^{x}}{e^{x}^{2}}

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Graph of f(x) = x^2 / exp(x) and its derivative-3-2-1123-150-100-5050100slope 0.3679f(x)f'(x)tangent at x = 1
Graph of f(x) = x^2 / exp(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 0.36788.

How to work it out

  1. 1

    We differentiate f(x)=x2exf(x) = \frac{x^{2}}{e^{x}} with respect to x. The rule that applies is the quotient rule.

    f(x)=x2exf(x) = \frac{x^{2}}{e^{x}}
  2. 2

    Quotient rule: (uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}. Let u=x2u = x^{2} and v=exv = e^{x}.

    u=x2,v=exu = x^{2},\quad v = e^{x}
  3. 3

    Differentiate each piece:

    u′=2x,v′=exu' = 2 x,\quad v' = e^{x}
  4. 4

    Put them in the rule:

    f′(x)=(2x)(ex)−(x2)(ex)(ex)2f'(x) = \frac{\left(2 x\right)\left(e^{x}\right) - \left(x^{2}\right)\left(e^{x}\right)}{\left(e^{x}\right)^{2}}
  5. 5

    Simplify:

    f'(x) = \frac{2 x e^{x} - x^{2} e^{x}}{e^{x}^{2}}
  6. 6

    Differentiating once more gives the second derivative, which tells you about curvature:

    f''(x) = \frac{\left(2 e^{x} + 2 x e^{x} - \left(2 x e^{x} + x^{2} e^{x}\right)\right) e^{x}^{2} - \left(2 x e^{x} - x^{2} e^{x}\right) 2 e^{x} e^{x}}{\left(e^{x}^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{x^{2}}{e^{x}}\right] = \frac{2 x e^{x} - x^{2} e^{x}}{e^{x}^{2}} \]
\[ f(x) = \frac{x^{2}}{e^{x}} \]
\[ u = x^{2},\quad v = e^{x} \]
\[ u' = 2 x,\quad v' = e^{x} \]
\[ f'(x) = \frac{\left(2 x\right)\left(e^{x}\right) - \left(x^{2}\right)\left(e^{x}\right)}{\left(e^{x}\right)^{2}} \]
\[ f'(x) = \frac{2 x e^{x} - x^{2} e^{x}}{e^{x}^{2}} \]
\[ f''(x) = \frac{\left(2 e^{x} + 2 x e^{x} - \left(2 x e^{x} + x^{2} e^{x}\right)\right) e^{x}^{2} - \left(2 x e^{x} - x^{2} e^{x}\right) 2 e^{x} e^{x}}{\left(e^{x}^{2}\right)^{2}} \]

Values of f and f' for x2x^{2} / exp(x)

xf(x)f'(x)
-229.556224-59.112449
-12.7182818-8.1548455
0.50.151632660.45489799
10.367879440.36787944
20.541341130
30.44808362-0.14936121

Questions

What is the derivative of x2x^{2} / exp(x)?

The derivative of x2ex\frac{x^{2}}{e^{x}} is \frac{2 x e^{x} - x^{2} e^{x}}{e^{x}^{2}}.

What is the second derivative of x2x^{2} / exp(x)?

((2exp(x) + 2x * exp(x) - (2x * exp(x) + x2x^{2} * exp(x))) * exp(x)^2 - (2x * exp(x) - x2x^{2} * exp(x)) * 2exp(x) * exp(x)) / (exp(x)^2)^2.

Which rule do you use for x2x^{2} / exp(x)?

The quotient rule.

What is the slope of x2x^{2} / exp(x) at x = 1?

0.36787944.

Related derivatives

    Derivative of x^2 / exp(x): (2x * exp(x) - x^2 * exp(x)) / exp(x)^2 (with steps) | WordGPT