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

The derivative of xsin⁡(x)\frac{\sqrt{x}}{\sin\left(x\right)} is sin⁡(x)2x−xcos⁡(x)sin⁡(x)2\frac{\frac{\sin\left(x\right)}{2 \sqrt{x}} - \sqrt{x} \cos\left(x\right)}{\sin\left(x\right)^{2}}.

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

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Graph of f(x) = sqrt(x) / sin(x) and its derivative-3-2-1123-2246810slope -0.1689f(x)f'(x)tangent at x = 1
Graph of f(x) = sqrt(x) / sin(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -0.16886.

How to work it out

  1. 1

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

    f(x)=xsin⁡(x)f(x) = \frac{\sqrt{x}}{\sin\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=xu = \sqrt{x} and v=sin⁡(x)v = \sin\left(x\right).

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

    Differentiate each piece:

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

    Put them in the rule:

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

    Simplify:

    f′(x)=sin⁡(x)2x−xcos⁡(x)sin⁡(x)2f'(x) = \frac{\frac{\sin\left(x\right)}{2 \sqrt{x}} - \sqrt{x} \cos\left(x\right)}{\sin\left(x\right)^{2}}
  6. 6

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

    f′′(x)=(cos⁡(x)2x−sin⁡(x)212x(2x)2−(cos⁡(x)2x+x−sin⁡(x)))sin⁡(x)2−(sin⁡(x)2x−xcos⁡(x))2sin⁡(x)cos⁡(x)(sin⁡(x)2)2f''(x) = \frac{\left(\frac{\cos\left(x\right) 2 \sqrt{x} - \sin\left(x\right) 2 \frac{1}{2 \sqrt{x}}}{\left(2 \sqrt{x}\right)^{2}} - \left(\frac{\cos\left(x\right)}{2 \sqrt{x}} + \sqrt{x} -\sin\left(x\right)\right)\right) \sin\left(x\right)^{2} - \left(\frac{\sin\left(x\right)}{2 \sqrt{x}} - \sqrt{x} \cos\left(x\right)\right) 2 \sin\left(x\right) \cos\left(x\right)}{\left(\sin\left(x\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\sqrt{x}}{\sin\left(x\right)}\right] = \frac{\frac{\sin\left(x\right)}{2 \sqrt{x}} - \sqrt{x} \cos\left(x\right)}{\sin\left(x\right)^{2}} \]
\[ f(x) = \frac{\sqrt{x}}{\sin\left(x\right)} \]
\[ u = \sqrt{x},\quad v = \sin\left(x\right) \]
\[ u' = \frac{1}{2 \sqrt{x}},\quad v' = \cos\left(x\right) \]
\[ f'(x) = \frac{\left(\frac{1}{2 \sqrt{x}}\right)\left(\sin\left(x\right)\right) - \left(\sqrt{x}\right)\left(\cos\left(x\right)\right)}{\left(\sin\left(x\right)\right)^{2}} \]
\[ f'(x) = \frac{\frac{\sin\left(x\right)}{2 \sqrt{x}} - \sqrt{x} \cos\left(x\right)}{\sin\left(x\right)^{2}} \]
\[ f''(x) = \frac{\left(\frac{\cos\left(x\right) 2 \sqrt{x} - \sin\left(x\right) 2 \frac{1}{2 \sqrt{x}}}{\left(2 \sqrt{x}\right)^{2}} - \left(\frac{\cos\left(x\right)}{2 \sqrt{x}} + \sqrt{x} -\sin\left(x\right)\right)\right) \sin\left(x\right)^{2} - \left(\frac{\sin\left(x\right)}{2 \sqrt{x}} - \sqrt{x} \cos\left(x\right)\right) 2 \sin\left(x\right) \cos\left(x\right)}{\left(\sin\left(x\right)^{2}\right)^{2}} \]

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

xf(x)f'(x)
0.51.4749043-1.2248899
11.1883951-0.16886217
21.55528161.1006068
312.27360288.148018

Questions

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

The derivative of xsin⁡(x)\frac{\sqrt{x}}{\sin\left(x\right)} is sin⁡(x)2x−xcos⁡(x)sin⁡(x)2\frac{\frac{\sin\left(x\right)}{2 \sqrt{x}} - \sqrt{x} \cos\left(x\right)}{\sin\left(x\right)^{2}}.

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

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

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

The quotient rule.

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

-0.16886217.

Related derivatives

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