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

The derivative of tan⁡(x)x\frac{\tan\left(x\right)}{\sqrt{x}} is xcos⁡(x)2−tan⁡(x)12xx2\frac{\frac{\sqrt{x}}{\cos\left(x\right)^{2}} - \tan\left(x\right) \frac{1}{2 \sqrt{x}}}{\sqrt{x}^{2}}.

f(x) =
ddx[tan⁡(x)x]=xcos⁡(x)2−tan⁡(x)12xx2\frac{d}{dx}\left[\frac{\tan\left(x\right)}{\sqrt{x}}\right] = \frac{\frac{\sqrt{x}}{\cos\left(x\right)^{2}} - \tan\left(x\right) \frac{1}{2 \sqrt{x}}}{\sqrt{x}^{2}}

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Graph of f(x) = tan(x) / sqrt(x) and its derivative-3-2-1123-10102030slope 2.647f(x)f'(x)tangent at x = 1
Graph of f(x) = tan(x) / sqrt(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 2.6468.

How to work it out

  1. 1

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

    f(x)=tan⁡(x)xf(x) = \frac{\tan\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=tan⁡(x)u = \tan\left(x\right) and v=xv = \sqrt{x}.

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

    Differentiate each piece:

    u′=1cos⁡(x)2,v′=12xu' = \frac{1}{\cos\left(x\right)^{2}},\quad v' = \frac{1}{2 \sqrt{x}}
  4. 4

    Put them in the rule:

    f′(x)=(1cos⁡(x)2)(x)−(tan⁡(x))(12x)(x)2f'(x) = \frac{\left(\frac{1}{\cos\left(x\right)^{2}}\right)\left(\sqrt{x}\right) - \left(\tan\left(x\right)\right)\left(\frac{1}{2 \sqrt{x}}\right)}{\left(\sqrt{x}\right)^{2}}
  5. 5

    Simplify:

    f′(x)=xcos⁡(x)2−tan⁡(x)12xx2f'(x) = \frac{\frac{\sqrt{x}}{\cos\left(x\right)^{2}} - \tan\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)=(cos⁡(x)22x−x2cos⁡(x)−sin⁡(x)(cos⁡(x)2)2−(12xcos⁡(x)2+tan⁡(x)−(212x)(2x)2))x2−(xcos⁡(x)2−tan⁡(x)12x)2x12x(x2)2f''(x) = \frac{\left(\frac{\frac{\cos\left(x\right)^{2}}{2 \sqrt{x}} - \sqrt{x} 2 \cos\left(x\right) -\sin\left(x\right)}{\left(\cos\left(x\right)^{2}\right)^{2}} - \left(\frac{\frac{1}{2 \sqrt{x}}}{\cos\left(x\right)^{2}} + \tan\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}}{\cos\left(x\right)^{2}} - \tan\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
\[ \frac{d}{dx}\left[\frac{\tan\left(x\right)}{\sqrt{x}}\right] = \frac{\frac{\sqrt{x}}{\cos\left(x\right)^{2}} - \tan\left(x\right) \frac{1}{2 \sqrt{x}}}{\sqrt{x}^{2}} \]
\[ f(x) = \frac{\tan\left(x\right)}{\sqrt{x}} \]
\[ u = \tan\left(x\right),\quad v = \sqrt{x} \]
\[ u' = \frac{1}{\cos\left(x\right)^{2}},\quad v' = \frac{1}{2 \sqrt{x}} \]
\[ f'(x) = \frac{\left(\frac{1}{\cos\left(x\right)^{2}}\right)\left(\sqrt{x}\right) - \left(\tan\left(x\right)\right)\left(\frac{1}{2 \sqrt{x}}\right)}{\left(\sqrt{x}\right)^{2}} \]
\[ f'(x) = \frac{\frac{\sqrt{x}}{\cos\left(x\right)^{2}} - \tan\left(x\right) \frac{1}{2 \sqrt{x}}}{\sqrt{x}^{2}} \]
\[ f''(x) = \frac{\left(\frac{\frac{\cos\left(x\right)^{2}}{2 \sqrt{x}} - \sqrt{x} 2 \cos\left(x\right) -\sin\left(x\right)}{\left(\cos\left(x\right)^{2}\right)^{2}} - \left(\frac{\frac{1}{2 \sqrt{x}}}{\cos\left(x\right)^{2}} + \tan\left(x\right) \frac{-\left(2 \frac{1}{2 \sqrt{x}}\right)}{\left(2 \sqrt{x}\right)^{2}}\right)\right) \sqrt{x}^{2} - \left(\frac{\sqrt{x}}{\cos\left(x\right)^{2}} - \tan\left(x\right) \frac{1}{2 \sqrt{x}}\right) 2 \sqrt{x} \frac{1}{2 \sqrt{x}}}{\left(\sqrt{x}^{2}\right)^{2}} \]

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

xf(x)f'(x)
0.50.772588391.0636921
11.55740772.646815
2-1.54505654.469381
3-0.0822992850.6027983

Questions

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

The derivative of tan⁡(x)x\frac{\tan\left(x\right)}{\sqrt{x}} is xcos⁡(x)2−tan⁡(x)12xx2\frac{\frac{\sqrt{x}}{\cos\left(x\right)^{2}} - \tan\left(x\right) \frac{1}{2 \sqrt{x}}}{\sqrt{x}^{2}}.

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

(((cos(x)^2 / (2sqrt(x)) - sqrt(x) * 2cos(x) * -sin(x)) / (cos(x)^2)^2 - (1 / (2sqrt(x)) / cos(x)^2 + tan(x) * -(2 * 1 / (2sqrt(x))) / (2sqrt(x))^2)) * sqrt(x)^2 - (sqrt(x) / cos(x)^2 - tan(x) * 1 / (2sqrt(x))) * 2sqrt(x) * 1 / (2sqrt(x))) / (sqrt(x)^2)^2.

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

The quotient rule.

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

2.646815.

Related derivatives

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