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Derivative of x3ex\frac{x^{3}}{e^{x}}

The derivative of x3ex\frac{x^{3}}{e^{x}} is \frac{3 x^{2} e^{x} - x^{3} e^{x}}{e^{x}^{2}}.

f(x) =
\frac{d}{dx}\left[\frac{x^{3}}{e^{x}}\right] = \frac{3 x^{2} e^{x} - x^{3} e^{x}}{e^{x}^{2}}

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Graph of f(x) = x^3 / exp(x) and its derivative-3-2-1123-200-100100200300400slope 0.7358f(x)f'(x)tangent at x = 1
Graph of f(x) = x^3 / exp(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 0.73576.

How to work it out

  1. 1

    We differentiate f(x)=x3exf(x) = \frac{x^{3}}{e^{x}} with respect to x. The rule that applies is the quotient rule.

    f(x)=x3exf(x) = \frac{x^{3}}{e^{x}}
  2. 2

    Quotient rule: (uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}. Let u=x3u = x^{3} and v=exv = e^{x}.

    u=x3,v=exu = x^{3},\quad v = e^{x}
  3. 3

    Differentiate each piece:

    u′=3x2,v′=exu' = 3 x^{2},\quad v' = e^{x}
  4. 4

    Put them in the rule:

    f′(x)=(3x2)(ex)−(x3)(ex)(ex)2f'(x) = \frac{\left(3 x^{2}\right)\left(e^{x}\right) - \left(x^{3}\right)\left(e^{x}\right)}{\left(e^{x}\right)^{2}}
  5. 5

    Simplify:

    f'(x) = \frac{3 x^{2} e^{x} - x^{3} e^{x}}{e^{x}^{2}}
  6. 6

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

    f''(x) = \frac{\left(6 x e^{x} + 3 x^{2} e^{x} - \left(3 x^{2} e^{x} + x^{3} e^{x}\right)\right) e^{x}^{2} - \left(3 x^{2} e^{x} - x^{3} e^{x}\right) 2 e^{x} e^{x}}{\left(e^{x}^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{x^{3}}{e^{x}}\right] = \frac{3 x^{2} e^{x} - x^{3} e^{x}}{e^{x}^{2}} \]
\[ f(x) = \frac{x^{3}}{e^{x}} \]
\[ u = x^{3},\quad v = e^{x} \]
\[ u' = 3 x^{2},\quad v' = e^{x} \]
\[ f'(x) = \frac{\left(3 x^{2}\right)\left(e^{x}\right) - \left(x^{3}\right)\left(e^{x}\right)}{\left(e^{x}\right)^{2}} \]
\[ f'(x) = \frac{3 x^{2} e^{x} - x^{3} e^{x}}{e^{x}^{2}} \]
\[ f''(x) = \frac{\left(6 x e^{x} + 3 x^{2} e^{x} - \left(3 x^{2} e^{x} + x^{3} e^{x}\right)\right) e^{x}^{2} - \left(3 x^{2} e^{x} - x^{3} e^{x}\right) 2 e^{x} e^{x}}{\left(e^{x}^{2}\right)^{2}} \]

Values of f and f' for x3x^{3} / exp(x)

xf(x)f'(x)
-2-59.112449147.78112
-1-2.718281810.873127
0.50.0758163320.37908166
10.367879440.73575888
21.08268230.54134113
31.34425080

Questions

What is the derivative of x3x^{3} / exp(x)?

The derivative of x3ex\frac{x^{3}}{e^{x}} is \frac{3 x^{2} e^{x} - x^{3} e^{x}}{e^{x}^{2}}.

What is the second derivative of x3x^{3} / exp(x)?

((6x * exp(x) + 3 * x2x^{2} * exp(x) - (3 * x2x^{2} * exp(x) + x3x^{3} * exp(x))) * exp(x)^2 - (3 * x2x^{2} * exp(x) - x3x^{3} * exp(x)) * 2exp(x) * exp(x)) / (exp(x)^2)^2.

Which rule do you use for x3x^{3} / exp(x)?

The quotient rule.

What is the slope of x3x^{3} / exp(x) at x = 1?

0.73575888.

Related derivatives

    Derivative of x^3 / exp(x): (3 * x^2 * exp(x) - x^3 * exp(x)) / exp(x)^2 (with steps) | WordGPT