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The derivative of the function/ sqrt(4r^2-x^2)

Derivative of sqrt(4r^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 
\/  4*r  - x
4r2−x2\sqrt{4 r^{2} - x^{2}} 
  /   ___________\
d |  /    2    2 |
--\\/  4*r  - x  /
dx
∂∂x4r2−x2\frac{\partial}{\partial x} \sqrt{4 r^{2} - x^{2}}
Transform
Detail solution

  1. Let u=4r2−x2u = 4 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(4r2−x2)\frac{\partial}{\partial x} \left(4 r^{2} - x^{2}\right):

    1. Differentiate 4r2−x24 r^{2} - x^{2} term by term:

      1. The derivative of the constant 4r24 r^{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:

    −x4r2−x2- \frac{x}{\sqrt{4 r^{2} - x^{2}}}

  4. Now simplify:

    −x4r2−x2- \frac{x}{\sqrt{4 r^{2} - x^{2}}}

The answer is:

−x4r2−x2- \frac{x}{\sqrt{4 r^{2} - x^{2}}}

The first derivative [src] 
     -x       
--------------
   ___________
  /    2    2 
\/  4*r  - x
−x4r2−x2- \frac{x}{\sqrt{4 r^{2} - x^{2}}}
Simplify
The second derivative [src] 
 /          2    \ 
 |         x     | 
-|1 + -----------| 
 |       2      2| 
 \    - x  + 4*r / 
-------------------
     _____________ 
    /    2      2  
  \/  - x  + 4*r
−x24r2−x2+14r2−x2- \frac{\frac{x^{2}}{4 r^{2} - x^{2}} + 1}{\sqrt{4 r^{2} - x^{2}}}
Simplify
The third derivative [src] 
     /          2    \
     |         x     |
-3*x*|1 + -----------|
     |       2      2|
     \    - x  + 4*r /
----------------------
                3/2   
   /   2      2\      
   \- x  + 4*r /
−3x(x24r2−x2+1)(4r2−x2)32- \frac{3 x \left(\frac{x^{2}}{4 r^{2} - x^{2}} + 1\right)}{\left(4 r^{2} - x^{2}\right)^{\frac{3}{2}}}
Simplify
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