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

The derivative of exarctan⁡(x)\frac{e^{x}}{\arctan\left(x\right)} is exarctan⁡(x)−ex11+x2arctan⁡(x)2\frac{e^{x} \arctan\left(x\right) - e^{x} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}}.

f(x) =
ddx[exarctan⁡(x)]=exarctan⁡(x)−ex11+x2arctan⁡(x)2\frac{d}{dx}\left[\frac{e^{x}}{\arctan\left(x\right)}\right] = \frac{e^{x} \arctan\left(x\right) - e^{x} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}}

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Graph of f(x) = exp(x) / atan(x) and its derivative-3-2-1123-15-10-551015slope 1.258f(x)f'(x)tangent at x = 1
Graph of f(x) = exp(x) / atan(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 1.2577.

How to work it out

  1. 1

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

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

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

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

    Differentiate each piece:

    u′=ex,v′=11+x2u' = e^{x},\quad v' = \frac{1}{1 + x^{2}}
  4. 4

    Put them in the rule:

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

    Simplify:

    f′(x)=exarctan⁡(x)−ex11+x2arctan⁡(x)2f'(x) = \frac{e^{x} \arctan\left(x\right) - e^{x} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}}
  6. 6

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

    f′′(x)=(exarctan⁡(x)+ex11+x2−(ex11+x2+ex−(2x)(1+x2)2))arctan⁡(x)2−(exarctan⁡(x)−ex11+x2)2arctan⁡(x)11+x2(arctan⁡(x)2)2f''(x) = \frac{\left(e^{x} \arctan\left(x\right) + e^{x} \frac{1}{1 + x^{2}} - \left(e^{x} \frac{1}{1 + x^{2}} + e^{x} \frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}}\right)\right) \arctan\left(x\right)^{2} - \left(e^{x} \arctan\left(x\right) - e^{x} \frac{1}{1 + x^{2}}\right) 2 \arctan\left(x\right) \frac{1}{1 + x^{2}}}{\left(\arctan\left(x\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{e^{x}}{\arctan\left(x\right)}\right] = \frac{e^{x} \arctan\left(x\right) - e^{x} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}} \]
\[ f(x) = \frac{e^{x}}{\arctan\left(x\right)} \]
\[ u = e^{x},\quad v = \arctan\left(x\right) \]
\[ u' = e^{x},\quad v' = \frac{1}{1 + x^{2}} \]
\[ f'(x) = \frac{\left(e^{x}\right)\left(\arctan\left(x\right)\right) - \left(e^{x}\right)\left(\frac{1}{1 + x^{2}}\right)}{\left(\arctan\left(x\right)\right)^{2}} \]
\[ f'(x) = \frac{e^{x} \arctan\left(x\right) - e^{x} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}} \]
\[ f''(x) = \frac{\left(e^{x} \arctan\left(x\right) + e^{x} \frac{1}{1 + x^{2}} - \left(e^{x} \frac{1}{1 + x^{2}} + e^{x} \frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}}\right)\right) \arctan\left(x\right)^{2} - \left(e^{x} \arctan\left(x\right) - e^{x} \frac{1}{1 + x^{2}}\right) 2 \arctan\left(x\right) \frac{1}{1 + x^{2}}}{\left(\arctan\left(x\right)^{2}\right)^{2}} \]

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

xf(x)f'(x)
-2-0.12223767-0.1443192
-1-0.46839865-0.7665905
0.53.5559792-2.5796793
13.46102391.2576677
26.67395085.4683403
316.08070514.793266

Questions

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

The derivative of exarctan⁡(x)\frac{e^{x}}{\arctan\left(x\right)} is exarctan⁡(x)−ex11+x2arctan⁡(x)2\frac{e^{x} \arctan\left(x\right) - e^{x} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}}.

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

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

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

The quotient rule.

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

1.2576677.

Related derivatives

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