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

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

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

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Graph of f(x) = sqrt(x) / cos(x) and its derivative-3-2-1123-20-101020304050slope 3.808f(x)f'(x)tangent at x = 1
Graph of f(x) = sqrt(x) / cos(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 3.8079.

How to work it out

  1. 1

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

    f(x)=xcos⁡(x)f(x) = \frac{\sqrt{x}}{\cos\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=cos⁡(x)v = \cos\left(x\right).

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

    Differentiate each piece:

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

    Put them in the rule:

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

    Simplify:

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

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

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

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

xf(x)f'(x)
0.50.805743881.2459238
11.85081573.8078826
2-3.39835236.5759472
3-1.7495595-0.042199592

Questions

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

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

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

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

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

The quotient rule.

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

3.8078826.

Related derivatives

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