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Derivative of x2\sqrt{x^{2}}

The derivative of x2\sqrt{x^{2}} is 2x2x2\frac{2 x}{2 \sqrt{x^{2}}}.

f(x) =
ddx[x2]=2x2x2\frac{d}{dx}\left[\sqrt{x^{2}}\right] = \frac{2 x}{2 \sqrt{x^{2}}}

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

How to work it out

  1. 1

    We differentiate f(x)=x2f(x) = \sqrt{x^{2}} with respect to x. The rule that applies is the chain rule.

    f(x)=x2f(x) = \sqrt{x^{2}}
  2. 2

    Chain rule: differentiate the outer function, keep the inside, then multiply by the derivative of the inside. The inside is g(x)=x2g(x) = x^{2}.

    g(x)=x2,gā€²(x)=2xg(x) = x^{2},\quad g'(x) = 2 x
  3. 3

    The outer function is sqrt(u); its derivative evaluated at g(x), times g'(x):

    fā€²(x)=[12u]u=g(x)ā‹…gā€²(x)f'(x) = \left[\frac{1}{2 \sqrt{u}}\right]_{u=g(x)} \cdot g'(x)
  4. 4

    Simplify:

    fā€²(x)=2x2x2f'(x) = \frac{2 x}{2 \sqrt{x^{2}}}
  5. 5

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

    fā€²ā€²(x)=4x2āˆ’2x22x2x2(2x2)2f''(x) = \frac{4 \sqrt{x^{2}} - 2 x 2 \frac{2 x}{2 \sqrt{x^{2}}}}{\left(2 \sqrt{x^{2}}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\sqrt{x^{2}}\right] = \frac{2 x}{2 \sqrt{x^{2}}} \]
\[ f(x) = \sqrt{x^{2}} \]
\[ g(x) = x^{2},\quad g'(x) = 2 x \]
\[ f'(x) = \left[\frac{1}{2 \sqrt{u}}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = \frac{2 x}{2 \sqrt{x^{2}}} \]
\[ f''(x) = \frac{4 \sqrt{x^{2}} - 2 x 2 \frac{2 x}{2 \sqrt{x^{2}}}}{\left(2 \sqrt{x^{2}}\right)^{2}} \]

Values of f and f' for sqrt(x2x^{2})

xf(x)f'(x)
-22-1
-11-1
0.50.51
111
221
331

Questions

What is the derivative of sqrt(x2x^{2})?

The derivative of x2\sqrt{x^{2}} is 2x2x2\frac{2 x}{2 \sqrt{x^{2}}}.

What is the second derivative of sqrt(x2x^{2})?

(4sqrt(x2x^{2}) - 2x * 2 * 2x / (2sqrt(x2x^{2}))) / (2sqrt(x2x^{2}))^2.

Which rule do you use for sqrt(x2x^{2})?

The chain rule.

What is the slope of sqrt(x2x^{2}) at x = 1?

1.

Related derivatives

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