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

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

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

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Graph of f(x) = cos(x) / sqrt(x) and its derivative-3-2-1123-224slope -1.112f(x)f'(x)tangent at x = 1
Graph of f(x) = cos(x) / sqrt(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -1.1116.

How to work it out

  1. 1

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

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

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

    Differentiate each piece:

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

    Put them in the rule:

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

    Simplify:

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

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

xf(x)f'(x)
0.51.2410892-1.9190993
10.54030231-1.1116221
2-0.29426025-0.56940531
3-0.571572430.013786398

Questions

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

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

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

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

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

The quotient rule.

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

-1.1116221.

Related derivatives

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