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Derivative of x\sqrt{x}

The derivative of x\sqrt{x} is 12x\frac{1}{2 \sqrt{x}}.

f(x) =
ddx[x]=12x\frac{d}{dx}\left[\sqrt{x}\right] = \frac{1}{2 \sqrt{x}}

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Graph of f(x) = sqrt(x) and its derivative-3-2-1123-1-0.50.511.52slope 0.5f(x)f'(x)tangent at x = 1
Graph of f(x) = sqrt(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 0.5.

How to work it out

  1. 1

    We differentiate f(x)=xf(x) = \sqrt{x} with respect to x. The rule that applies is the standard derivative.

    f(x)=xf(x) = \sqrt{x}
  2. 2

    This is a standard derivative from the table.

  3. 3

    Simplify:

    fā€²(x)=12xf'(x) = \frac{1}{2 \sqrt{x}}
  4. 4

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

    fā€²ā€²(x)=āˆ’(212x)(2x)2f''(x) = \frac{-\left(2 \frac{1}{2 \sqrt{x}}\right)}{\left(2 \sqrt{x}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\sqrt{x}\right] = \frac{1}{2 \sqrt{x}} \]
\[ f(x) = \sqrt{x} \]
\[ f'(x) = \frac{1}{2 \sqrt{x}} \]
\[ f''(x) = \frac{-\left(2 \frac{1}{2 \sqrt{x}}\right)}{\left(2 \sqrt{x}\right)^{2}} \]

Values of f and f' for sqrt(x)

xf(x)f'(x)
0.50.707106780.70710678
110.5
21.41421360.35355339
31.73205080.28867513

Questions

What is the derivative of sqrt(x)?

The derivative of x\sqrt{x} is 12x\frac{1}{2 \sqrt{x}}.

What is the second derivative of sqrt(x)?

-(2 * 1 / (2sqrt(x))) / (2sqrt(x))^2.

Which rule do you use for sqrt(x)?

The standard derivative.

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

0.5.

Related derivatives

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