Derivative of √(1-x²)

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   ________
  /      2 
\/  1 - x

$$\sqrt{- x^{2} + 1}$$

  /   ________\
d |  /      2 |
--\\/  1 - x  /
dx

$$\frac{d}{d x} \sqrt{- x^{2} + 1}$$

Transform

Detail solution

Let $u = 1 - x^{2}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - x^{2}\right)$ :
1. Differentiate $1 - x^{2}$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $- 2 x$
  The result is: $- 2 x$
The result of the chain rule is:

$- \frac{x}{\sqrt{1 - x^{2}}}$

The answer is:

$- \frac{x}{\sqrt{1 - x^{2}}}$

The graph

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The first derivative [src]

    -x     
-----------
   ________
  /      2 
\/  1 - x

$$- \frac{x}{\sqrt{- x^{2} + 1}}$$

Simplify

The second derivative [src]

 /       2  \ 
 |      x   | 
-|1 + ------| 
 |         2| 
 \    1 - x / 
--------------
    ________  
   /      2   
 \/  1 - x

$$- \frac{\frac{x^{2}}{- x^{2} + 1} + 1}{\sqrt{- x^{2} + 1}}$$

Simplify

The third derivative [src]

     /       2  \
     |      x   |
-3*x*|1 + ------|
     |         2|
     \    1 - x /
-----------------
           3/2   
   /     2\      
   \1 - x /

$$- \frac{3 x \left(\frac{x^{2}}{- x^{2} + 1} + 1\right)}{\left(- x^{2} + 1\right)^{\frac{3}{2}}}$$

Simplify

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Derivative of √(1-x²)

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