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Derivative of arctan⁡(x)x\frac{\arctan\left(x\right)}{\sqrt{x}}

The derivative of arctan⁡(x)x\frac{\arctan\left(x\right)}{\sqrt{x}} is x1+x2−arctan⁡(x)12xx2\frac{\frac{\sqrt{x}}{1 + x^{2}} - \arctan\left(x\right) \frac{1}{2 \sqrt{x}}}{\sqrt{x}^{2}}.

f(x) =
ddx[arctan⁡(x)x]=x1+x2−arctan⁡(x)12xx2\frac{d}{dx}\left[\frac{\arctan\left(x\right)}{\sqrt{x}}\right] = \frac{\frac{\sqrt{x}}{1 + x^{2}} - \arctan\left(x\right) \frac{1}{2 \sqrt{x}}}{\sqrt{x}^{2}}

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Graph of f(x) = atan(x) / sqrt(x) and its derivative-3-2-1123-0.20.20.40.60.81slope 0.1073f(x)f'(x)tangent at x = 1
Graph of f(x) = atan(x) / sqrt(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 0.1073.

How to work it out

  1. 1

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

    f(x)=arctan⁡(x)xf(x) = \frac{\arctan\left(x\right)}{\sqrt{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=xv = \sqrt{x}.

    u=arctan⁡(x),v=xu = \arctan\left(x\right),\quad v = \sqrt{x}
  3. 3

    Differentiate each piece:

    u′=11+x2,v′=12xu' = \frac{1}{1 + x^{2}},\quad v' = \frac{1}{2 \sqrt{x}}
  4. 4

    Put them in the rule:

    f′(x)=(11+x2)(x)−(arctan⁡(x))(12x)(x)2f'(x) = \frac{\left(\frac{1}{1 + x^{2}}\right)\left(\sqrt{x}\right) - \left(\arctan\left(x\right)\right)\left(\frac{1}{2 \sqrt{x}}\right)}{\left(\sqrt{x}\right)^{2}}
  5. 5

    Simplify:

    f′(x)=x1+x2−arctan⁡(x)12xx2f'(x) = \frac{\frac{\sqrt{x}}{1 + x^{2}} - \arctan\left(x\right) \frac{1}{2 \sqrt{x}}}{\sqrt{x}^{2}}
  6. 6

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

    f′′(x)=(1+x22x−x2x(1+x2)2−(12x1+x2+arctan⁡(x)−(212x)(2x)2))x2−(x1+x2−arctan⁡(x)12x)2x12x(x2)2f''(x) = \frac{\left(\frac{\frac{1 + x^{2}}{2 \sqrt{x}} - \sqrt{x} 2 x}{\left(1 + x^{2}\right)^{2}} - \left(\frac{\frac{1}{2 \sqrt{x}}}{1 + x^{2}} + \arctan\left(x\right) \frac{-\left(2 \frac{1}{2 \sqrt{x}}\right)}{\left(2 \sqrt{x}\right)^{2}}\right)\right) \sqrt{x}^{2} - \left(\frac{\sqrt{x}}{1 + x^{2}} - \arctan\left(x\right) \frac{1}{2 \sqrt{x}}\right) 2 \sqrt{x} \frac{1}{2 \sqrt{x}}}{\left(\sqrt{x}^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\arctan\left(x\right)}{\sqrt{x}}\right] = \frac{\frac{\sqrt{x}}{1 + x^{2}} - \arctan\left(x\right) \frac{1}{2 \sqrt{x}}}{\sqrt{x}^{2}} \]
\[ f(x) = \frac{\arctan\left(x\right)}{\sqrt{x}} \]
\[ u = \arctan\left(x\right),\quad v = \sqrt{x} \]
\[ u' = \frac{1}{1 + x^{2}},\quad v' = \frac{1}{2 \sqrt{x}} \]
\[ f'(x) = \frac{\left(\frac{1}{1 + x^{2}}\right)\left(\sqrt{x}\right) - \left(\arctan\left(x\right)\right)\left(\frac{1}{2 \sqrt{x}}\right)}{\left(\sqrt{x}\right)^{2}} \]
\[ f'(x) = \frac{\frac{\sqrt{x}}{1 + x^{2}} - \arctan\left(x\right) \frac{1}{2 \sqrt{x}}}{\sqrt{x}^{2}} \]
\[ f''(x) = \frac{\left(\frac{\frac{1 + x^{2}}{2 \sqrt{x}} - \sqrt{x} 2 x}{\left(1 + x^{2}\right)^{2}} - \left(\frac{\frac{1}{2 \sqrt{x}}}{1 + x^{2}} + \arctan\left(x\right) \frac{-\left(2 \frac{1}{2 \sqrt{x}}\right)}{\left(2 \sqrt{x}\right)^{2}}\right)\right) \sqrt{x}^{2} - \left(\frac{\sqrt{x}}{1 + x^{2}} - \arctan\left(x\right) \frac{1}{2 \sqrt{x}}\right) 2 \sqrt{x} \frac{1}{2 \sqrt{x}}}{\left(\sqrt{x}^{2}\right)^{2}} \]

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

xf(x)f'(x)
0.50.655696740.47567411
10.785398160.10730092
20.78287237-0.054296735
30.72113691-0.062454459

Questions

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

The derivative of arctan⁡(x)x\frac{\arctan\left(x\right)}{\sqrt{x}} is x1+x2−arctan⁡(x)12xx2\frac{\frac{\sqrt{x}}{1 + x^{2}} - \arctan\left(x\right) \frac{1}{2 \sqrt{x}}}{\sqrt{x}^{2}}.

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

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

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

The quotient rule.

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

0.10730092.

Related derivatives

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