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

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

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

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Graph of f(x) = sqrt(x) * cos(x) and its derivative-3-2-1123-2-1123slope -0.5713f(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) = -0.57132.

How to work it out

  1. 1

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

    f(x)=xcos⁡(x)f(x) = \sqrt{x} \cos\left(x\right)
  2. 2

    Product rule: (uv)′=u′v+uv′(uv)' = u'v + uv'. 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))f'(x) = \left(\frac{1}{2 \sqrt{x}}\right)\left(\cos\left(x\right)\right) + \left(\sqrt{x}\right)\left(-\sin\left(x\right)\right)
  5. 5

    Simplify:

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

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

xf(x)f'(x)
0.50.620544580.28153953
10.54030231-0.57131983
2-0.5885205-1.4330709
3-1.7147173-0.53021324

Questions

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

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

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).

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

The product rule.

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

-0.57131983.

Related derivatives

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