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Derivative of x1x\frac{\sqrt{x}}{\frac{1}{x}}

The derivative of x1x\frac{\sqrt{x}}{\frac{1}{x}} is 1x2xāˆ’xāˆ’1x21x2\frac{\frac{\frac{1}{x}}{2 \sqrt{x}} - \sqrt{x} \frac{-1}{x^{2}}}{\frac{1}{x}^{2}}.

f(x) =
ddx[x1x]=1x2xāˆ’xāˆ’1x21x2\frac{d}{dx}\left[\frac{\sqrt{x}}{\frac{1}{x}}\right] = \frac{\frac{\frac{1}{x}}{2 \sqrt{x}} - \sqrt{x} \frac{-1}{x^{2}}}{\frac{1}{x}^{2}}

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Graph of f(x) = sqrt(x) / (1 / x) and its derivative-3-2-1123-4-224slope 1.5f(x)f'(x)tangent at x = 1
Graph of f(x) = sqrt(x) / (1 / x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 1.5.

How to work it out

  1. 1

    We differentiate f(x)=x1xf(x) = \frac{\sqrt{x}}{\frac{1}{x}} with respect to x. The rule that applies is the quotient rule.

    f(x)=x1xf(x) = \frac{\sqrt{x}}{\frac{1}{x}}
  2. 2

    Quotient rule: (uv)ā€²=uā€²vāˆ’uvā€²v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}. Let u=xu = \sqrt{x} and v=1xv = \frac{1}{x}.

    u=x,v=1xu = \sqrt{x},\quad v = \frac{1}{x}
  3. 3

    Differentiate each piece:

    uā€²=12x,vā€²=āˆ’1x2u' = \frac{1}{2 \sqrt{x}},\quad v' = \frac{-1}{x^{2}}
  4. 4

    Put them in the rule:

    fā€²(x)=(12x)(1x)āˆ’(x)(āˆ’1x2)(1x)2f'(x) = \frac{\left(\frac{1}{2 \sqrt{x}}\right)\left(\frac{1}{x}\right) - \left(\sqrt{x}\right)\left(\frac{-1}{x^{2}}\right)}{\left(\frac{1}{x}\right)^{2}}
  5. 5

    Simplify:

    fā€²(x)=1x2xāˆ’xāˆ’1x21x2f'(x) = \frac{\frac{\frac{1}{x}}{2 \sqrt{x}} - \sqrt{x} \frac{-1}{x^{2}}}{\frac{1}{x}^{2}}
  6. 6

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

    fā€²ā€²(x)=(āˆ’1x22xāˆ’212xx(2x)2āˆ’(āˆ’1x22x+xāˆ’((āˆ’2)x)(x2)2))1x2āˆ’(1x2xāˆ’xāˆ’1x2)21xāˆ’1x2(1x2)2f''(x) = \frac{\left(\frac{\frac{-1}{x^{2}} 2 \sqrt{x} - \frac{2 \frac{1}{2 \sqrt{x}}}{x}}{\left(2 \sqrt{x}\right)^{2}} - \left(\frac{\frac{-1}{x^{2}}}{2 \sqrt{x}} + \sqrt{x} \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}\right)\right) \frac{1}{x}^{2} - \left(\frac{\frac{1}{x}}{2 \sqrt{x}} - \sqrt{x} \frac{-1}{x^{2}}\right) 2 \frac{1}{x} \frac{-1}{x^{2}}}{\left(\frac{1}{x}^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\sqrt{x}}{\frac{1}{x}}\right] = \frac{\frac{\frac{1}{x}}{2 \sqrt{x}} - \sqrt{x} \frac{-1}{x^{2}}}{\frac{1}{x}^{2}} \]
\[ f(x) = \frac{\sqrt{x}}{\frac{1}{x}} \]
\[ u = \sqrt{x},\quad v = \frac{1}{x} \]
\[ u' = \frac{1}{2 \sqrt{x}},\quad v' = \frac{-1}{x^{2}} \]
\[ f'(x) = \frac{\left(\frac{1}{2 \sqrt{x}}\right)\left(\frac{1}{x}\right) - \left(\sqrt{x}\right)\left(\frac{-1}{x^{2}}\right)}{\left(\frac{1}{x}\right)^{2}} \]
\[ f'(x) = \frac{\frac{\frac{1}{x}}{2 \sqrt{x}} - \sqrt{x} \frac{-1}{x^{2}}}{\frac{1}{x}^{2}} \]
\[ f''(x) = \frac{\left(\frac{\frac{-1}{x^{2}} 2 \sqrt{x} - \frac{2 \frac{1}{2 \sqrt{x}}}{x}}{\left(2 \sqrt{x}\right)^{2}} - \left(\frac{\frac{-1}{x^{2}}}{2 \sqrt{x}} + \sqrt{x} \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}\right)\right) \frac{1}{x}^{2} - \left(\frac{\frac{1}{x}}{2 \sqrt{x}} - \sqrt{x} \frac{-1}{x^{2}}\right) 2 \frac{1}{x} \frac{-1}{x^{2}}}{\left(\frac{1}{x}^{2}\right)^{2}} \]

Values of f and f' for sqrt(x) / (1 / x)

xf(x)f'(x)
0.50.353553391.0606602
111.5
22.82842712.1213203
35.19615242.5980762

Questions

What is the derivative of sqrt(x) / (1 / x)?

The derivative of x1x\frac{\sqrt{x}}{\frac{1}{x}} is 1x2xāˆ’xāˆ’1x21x2\frac{\frac{\frac{1}{x}}{2 \sqrt{x}} - \sqrt{x} \frac{-1}{x^{2}}}{\frac{1}{x}^{2}}.

What is the second derivative of sqrt(x) / (1 / x)?

(((-1 / x2x^{2} * 2sqrt(x) - 2 * 1 / (2sqrt(x)) / x) / (2sqrt(x))^2 - (-1 / x2x^{2} / (2sqrt(x)) + sqrt(x) * -(-2x) / (x2x^{2})^2)) * (1 / x)^2 - (1 / x / (2sqrt(x)) - sqrt(x) * -1 / x2x^{2}) * 2 * 1 / x * -1 / x2x^{2}) / ((1 / x)^2)^2.

Which rule do you use for sqrt(x) / (1 / x)?

The quotient rule.

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

1.5.

Related derivatives

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