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Derivative of x3exx^{3} e^{x}

The derivative of x3exx^{3} e^{x} is 3x2ex+x3ex3 x^{2} e^{x} + x^{3} e^{x}.

f(x) =
ddx[x3ex]=3x2ex+x3ex\frac{d}{dx}\left[x^{3} e^{x}\right] = 3 x^{2} e^{x} + x^{3} e^{x}

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Graph of f(x) = x^3 * exp(x) and its derivative-3-2-1123-100100200300400500slope 10.87f(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) = 10.873.

How to work it out

  1. 1

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

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

    Product rule: (uv)′=u′v+uv′(uv)' = u'v + uv'. 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)f'(x) = \left(3 x^{2}\right)\left(e^{x}\right) + \left(x^{3}\right)\left(e^{x}\right)
  5. 5

    Simplify:

    f′(x)=3x2ex+x3exf'(x) = 3 x^{2} e^{x} + x^{3} e^{x}
  6. 6

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

    f′′(x)=6xex+3x2ex+3x2ex+x3exf''(x) = 6 x e^{x} + 3 x^{2} e^{x} + 3 x^{2} e^{x} + x^{3} e^{x}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[x^{3} e^{x}\right] = 3 x^{2} e^{x} + x^{3} e^{x} \]
\[ f(x) = x^{3} e^{x} \]
\[ u = x^{3},\quad v = e^{x} \]
\[ u' = 3 x^{2},\quad v' = e^{x} \]
\[ f'(x) = \left(3 x^{2}\right)\left(e^{x}\right) + \left(x^{3}\right)\left(e^{x}\right) \]
\[ f'(x) = 3 x^{2} e^{x} + x^{3} e^{x} \]
\[ f''(x) = 6 x e^{x} + 3 x^{2} e^{x} + 3 x^{2} e^{x} + x^{3} e^{x} \]

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

xf(x)f'(x)
-2-1.08268230.54134113
-1-0.367879440.73575888
0.50.206090161.4426311
12.718281810.873127
259.112449147.78112
3542.30951084.619

Questions

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

The derivative of x3exx^{3} e^{x} is 3x2ex+x3ex3 x^{2} e^{x} + x^{3} e^{x}.

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).

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

The product rule.

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

10.873127.

Related derivatives

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