WordGPT

Derivative of x4ex\frac{x^{4}}{e^{x}}

The derivative of x4ex\frac{x^{4}}{e^{x}} is \frac{4 x^{3} e^{x} - x^{4} e^{x}}{e^{x}^{2}}.

f(x) =
\frac{d}{dx}\left[\frac{x^{4}}{e^{x}}\right] = \frac{4 x^{3} e^{x} - x^{4} e^{x}}{e^{x}^{2}}

Type any function of x; the derivative is computed symbolically in your browser.

Explain it step by step in chatMake a short video of it
Graph of f(x) = x^4 / exp(x) and its derivative-3-2-1123-1200-1000-800-600-400-200200400600slope 1.104f(x)f'(x)tangent at x = 1
Graph of f(x) = x^4 / exp(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 1.1036.

How to work it out

  1. 1

    We differentiate f(x)=x4exf(x) = \frac{x^{4}}{e^{x}} with respect to x. The rule that applies is the quotient rule.

    f(x)=x4exf(x) = \frac{x^{4}}{e^{x}}
  2. 2

    Quotient rule: (uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}. Let u=x4u = x^{4} and v=exv = e^{x}.

    u=x4,v=exu = x^{4},\quad v = e^{x}
  3. 3

    Differentiate each piece:

    u′=4x3,v′=exu' = 4 x^{3},\quad v' = e^{x}
  4. 4

    Put them in the rule:

    f′(x)=(4x3)(ex)−(x4)(ex)(ex)2f'(x) = \frac{\left(4 x^{3}\right)\left(e^{x}\right) - \left(x^{4}\right)\left(e^{x}\right)}{\left(e^{x}\right)^{2}}
  5. 5

    Simplify:

    f'(x) = \frac{4 x^{3} e^{x} - x^{4} e^{x}}{e^{x}^{2}}
  6. 6

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

    f''(x) = \frac{\left(12 x^{2} e^{x} + 4 x^{3} e^{x} - \left(4 x^{3} e^{x} + x^{4} e^{x}\right)\right) e^{x}^{2} - \left(4 x^{3} e^{x} - x^{4} e^{x}\right) 2 e^{x} e^{x}}{\left(e^{x}^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{x^{4}}{e^{x}}\right] = \frac{4 x^{3} e^{x} - x^{4} e^{x}}{e^{x}^{2}} \]
\[ f(x) = \frac{x^{4}}{e^{x}} \]
\[ u = x^{4},\quad v = e^{x} \]
\[ u' = 4 x^{3},\quad v' = e^{x} \]
\[ f'(x) = \frac{\left(4 x^{3}\right)\left(e^{x}\right) - \left(x^{4}\right)\left(e^{x}\right)}{\left(e^{x}\right)^{2}} \]
\[ f'(x) = \frac{4 x^{3} e^{x} - x^{4} e^{x}}{e^{x}^{2}} \]
\[ f''(x) = \frac{\left(12 x^{2} e^{x} + 4 x^{3} e^{x} - \left(4 x^{3} e^{x} + x^{4} e^{x}\right)\right) e^{x}^{2} - \left(4 x^{3} e^{x} - x^{4} e^{x}\right) 2 e^{x} e^{x}}{\left(e^{x}^{2}\right)^{2}} \]

Values of f and f' for x4x^{4} / exp(x)

xf(x)f'(x)
-2118.2249-354.67469
-12.7182818-13.591409
0.50.0379081660.26535716
10.367879441.1036383
22.16536452.1653645
34.03275251.3442508

Questions

What is the derivative of x4x^{4} / exp(x)?

The derivative of x4ex\frac{x^{4}}{e^{x}} is \frac{4 x^{3} e^{x} - x^{4} e^{x}}{e^{x}^{2}}.

What is the second derivative of x4x^{4} / exp(x)?

((12 * x2x^{2} * exp(x) + 4 * x3x^{3} * exp(x) - (4 * x3x^{3} * exp(x) + x4x^{4} * exp(x))) * exp(x)^2 - (4 * x3x^{3} * exp(x) - x4x^{4} * exp(x)) * 2exp(x) * exp(x)) / (exp(x)^2)^2.

Which rule do you use for x4x^{4} / exp(x)?

The quotient rule.

What is the slope of x4x^{4} / exp(x) at x = 1?

1.1036383.

Related derivatives

    Derivative of x^4 / exp(x): (4 * x^3 * exp(x) - x^4 * exp(x)) / exp(x)^2 (with steps) | WordGPT