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Derivative of 1āˆ’x\sqrt{1 - x}

The derivative of 1āˆ’x\sqrt{1 - x} is āˆ’121āˆ’x\frac{-1}{2 \sqrt{1 - x}}.

f(x) =
ddx[1āˆ’x]=āˆ’121āˆ’x\frac{d}{dx}\left[\sqrt{1 - x}\right] = \frac{-1}{2 \sqrt{1 - x}}

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Graph of f(x) = sqrt(1 - x) and its derivative-3-2-1123-1-0.50.511.52slope -0.2887f(x)f'(x)tangent at x = -2
Graph of f(x) = sqrt(1 - x) and its derivative. The dashed line is the tangent at x = -2; its slope is f'(-2) = -0.28868.

How to work it out

  1. 1

    We differentiate f(x)=1āˆ’xf(x) = \sqrt{1 - x} with respect to x. The rule that applies is the chain rule.

    f(x)=1āˆ’xf(x) = \sqrt{1 - x}
  2. 2

    Chain rule: differentiate the outer function, keep the inside, then multiply by the derivative of the inside. The inside is g(x)=1āˆ’xg(x) = 1 - x.

    g(x)=1āˆ’x,gā€²(x)=āˆ’1g(x) = 1 - x,\quad g'(x) = -1
  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)=āˆ’121āˆ’xf'(x) = \frac{-1}{2 \sqrt{1 - x}}
  5. 5

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

    fā€²ā€²(x)=āˆ’((āˆ’2)āˆ’121āˆ’x)(21āˆ’x)2f''(x) = \frac{-\left(\left(-2\right) \frac{-1}{2 \sqrt{1 - x}}\right)}{\left(2 \sqrt{1 - x}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\sqrt{1 - x}\right] = \frac{-1}{2 \sqrt{1 - x}} \]
\[ f(x) = \sqrt{1 - x} \]
\[ g(x) = 1 - x,\quad g'(x) = -1 \]
\[ f'(x) = \left[\frac{1}{2 \sqrt{u}}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = \frac{-1}{2 \sqrt{1 - x}} \]
\[ f''(x) = \frac{-\left(\left(-2\right) \frac{-1}{2 \sqrt{1 - x}}\right)}{\left(2 \sqrt{1 - x}\right)^{2}} \]

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

xf(x)f'(x)
-21.7320508-0.28867513
-11.4142136-0.35355339
0.50.70710678-0.70710678

Questions

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

The derivative of 1āˆ’x\sqrt{1 - x} is āˆ’121āˆ’x\frac{-1}{2 \sqrt{1 - x}}.

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

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

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

The chain rule.

What is the slope of sqrt(1 - x) at x = -2?

-0.28867513.

Related derivatives

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