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

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

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

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Graph of f(x) = sin(x) / sqrt(x) and its derivative-3-2-1123-0.50.51slope 0.1196f(x)f'(x)tangent at x = 1
Graph of f(x) = sin(x) / sqrt(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 0.11957.

How to work it out

  1. 1

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

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

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

    Differentiate each piece:

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

    Put them in the rule:

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

    Simplify:

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

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

xf(x)f'(x)
0.50.67801010.56307906
10.841470980.11956681
20.64297038-0.45500284
30.081475675-0.58515171

Questions

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

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

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

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

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

The quotient rule.

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

0.11956681.

Related derivatives

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