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The derivative of the function/ sqrt(r^2-x^2)

Derivative of sqrt(r^2-x^2)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   _________
  /  2    2 
\/  r  - x
r2−x2\sqrt{r^{2} - x^{2}} 
  /   _________\
d |  /  2    2 |
--\\/  r  - x  /
dx
∂∂xr2−x2\frac{\partial}{\partial x} \sqrt{r^{2} - x^{2}}
Transform
Detail solution

  1. Let u=r2−x2u = r^{2} - x^{2}.

  2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

  3. Then, apply the chain rule. Multiply by ∂∂x(r2−x2)\frac{\partial}{\partial x} \left(r^{2} - x^{2}\right):

    1. Differentiate r2−x2r^{2} - x^{2} term by term:

      1. The derivative of the constant r2r^{2} 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: x2x^{2} goes to 2x2 x

        So, the result is: −2x- 2 x

      The result is: −2x- 2 x

    The result of the chain rule is:

    −xr2−x2- \frac{x}{\sqrt{r^{2} - x^{2}}}

The answer is:

−xr2−x2- \frac{x}{\sqrt{r^{2} - x^{2}}}

The first derivative [src] 
    -x      
------------
   _________
  /  2    2 
\/  r  - x
−xr2−x2- \frac{x}{\sqrt{r^{2} - x^{2}}}
Simplify
The second derivative [src] 
 /        2  \ 
 |       x   | 
-|1 + -------| 
 |     2    2| 
 \    r  - x / 
---------------
     _________ 
    /  2    2  
  \/  r  - x
−x2r2−x2+1r2−x2- \frac{\frac{x^{2}}{r^{2} - x^{2}} + 1}{\sqrt{r^{2} - x^{2}}}
Simplify
The third derivative [src] 
     /        2  \
     |       x   |
-3*x*|1 + -------|
     |     2    2|
     \    r  - x /
------------------
            3/2   
   / 2    2\      
   \r  - x /
−3x(x2r2−x2+1)(r2−x2)32- \frac{3 x \left(\frac{x^{2}}{r^{2} - x^{2}} + 1\right)}{\left(r^{2} - x^{2}\right)^{\frac{3}{2}}}
Simplify
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