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Derivative of arctan⁡(x)ex\frac{\arctan\left(x\right)}{e^{x}}

The derivative of arctan⁡(x)ex\frac{\arctan\left(x\right)}{e^{x}} is \frac{\frac{e^{x}}{1 + x^{2}} - \arctan\left(x\right) e^{x}}{e^{x}^{2}}.

f(x) =
\frac{d}{dx}\left[\frac{\arctan\left(x\right)}{e^{x}}\right] = \frac{\frac{e^{x}}{1 + x^{2}} - \arctan\left(x\right) e^{x}}{e^{x}^{2}}

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Graph of f(x) = atan(x) / exp(x) and its derivative-3-2-1123-15-10-551015slope -0.105f(x)f'(x)tangent at x = 1
Graph of f(x) = atan(x) / exp(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -0.10499.

How to work it out

  1. 1

    We differentiate f(x)=arctan⁡(x)exf(x) = \frac{\arctan\left(x\right)}{e^{x}} with respect to x. The rule that applies is the quotient rule.

    f(x)=arctan⁡(x)exf(x) = \frac{\arctan\left(x\right)}{e^{x}}
  2. 2

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

    u=arctan⁡(x),v=exu = \arctan\left(x\right),\quad v = e^{x}
  3. 3

    Differentiate each piece:

    u′=11+x2,v′=exu' = \frac{1}{1 + x^{2}},\quad v' = e^{x}
  4. 4

    Put them in the rule:

    f′(x)=(11+x2)(ex)−(arctan⁡(x))(ex)(ex)2f'(x) = \frac{\left(\frac{1}{1 + x^{2}}\right)\left(e^{x}\right) - \left(\arctan\left(x\right)\right)\left(e^{x}\right)}{\left(e^{x}\right)^{2}}
  5. 5

    Simplify:

    f'(x) = \frac{\frac{e^{x}}{1 + x^{2}} - \arctan\left(x\right) e^{x}}{e^{x}^{2}}
  6. 6

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

    f''(x) = \frac{\left(\frac{e^{x} \left(1 + x^{2}\right) - e^{x} 2 x}{\left(1 + x^{2}\right)^{2}} - \left(\frac{e^{x}}{1 + x^{2}} + \arctan\left(x\right) e^{x}\right)\right) e^{x}^{2} - \left(\frac{e^{x}}{1 + x^{2}} - \arctan\left(x\right) e^{x}\right) 2 e^{x} e^{x}}{\left(e^{x}^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\arctan\left(x\right)}{e^{x}}\right] = \frac{\frac{e^{x}}{1 + x^{2}} - \arctan\left(x\right) e^{x}}{e^{x}^{2}} \]
\[ f(x) = \frac{\arctan\left(x\right)}{e^{x}} \]
\[ u = \arctan\left(x\right),\quad v = e^{x} \]
\[ u' = \frac{1}{1 + x^{2}},\quad v' = e^{x} \]
\[ f'(x) = \frac{\left(\frac{1}{1 + x^{2}}\right)\left(e^{x}\right) - \left(\arctan\left(x\right)\right)\left(e^{x}\right)}{\left(e^{x}\right)^{2}} \]
\[ f'(x) = \frac{\frac{e^{x}}{1 + x^{2}} - \arctan\left(x\right) e^{x}}{e^{x}^{2}} \]
\[ f''(x) = \frac{\left(\frac{e^{x} \left(1 + x^{2}\right) - e^{x} 2 x}{\left(1 + x^{2}\right)^{2}} - \left(\frac{e^{x}}{1 + x^{2}} + \arctan\left(x\right) e^{x}\right)\right) e^{x}^{2} - \left(\frac{e^{x}}{1 + x^{2}} - \arctan\left(x\right) e^{x}\right) 2 e^{x} e^{x}}{\left(e^{x}^{2}\right)^{2}} \]

Values of f and f' for atan(x) / exp(x)

xf(x)f'(x)
-2-8.1807849.6585952
-1-2.13493363.4940745
0.50.281216490.20400804
10.28893184-0.10499212
20.14983629-0.12276923
30.062186327-0.05720762

Questions

What is the derivative of atan(x) / exp(x)?

The derivative of arctan⁡(x)ex\frac{\arctan\left(x\right)}{e^{x}} is \frac{\frac{e^{x}}{1 + x^{2}} - \arctan\left(x\right) e^{x}}{e^{x}^{2}}.

What is the second derivative of atan(x) / exp(x)?

(((exp(x) * (1 + x2x^{2}) - exp(x) * 2x) / (1 + x2x^{2})^2 - (exp(x) / (1 + x2x^{2}) + atan(x) * exp(x))) * exp(x)^2 - (exp(x) / (1 + x2x^{2}) - atan(x) * exp(x)) * 2exp(x) * exp(x)) / (exp(x)^2)^2.

Which rule do you use for atan(x) / exp(x)?

The quotient rule.

What is the slope of atan(x) / exp(x) at x = 1?

-0.10499212.

Related derivatives

    Derivative of atan(x) / exp(x): (exp(x) / (1 + x^2) - atan(x) * exp(x)) / exp(x)^2 (with steps) | WordGPT