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Derivative of:
Derivative of x^-3
Derivative of x^2*sqrt(x)
Derivative of 1/2*x^2
Derivative of x*(log(x)/log(10))
Identical expressions
(x^ two)*(sqrt(five -x^ two))
(x squared ) multiply by ( square root of (5 minus x squared ))
(x to the power of two) multiply by ( square root of (five minus x to the power of two))
(x^2)*(√(5-x^2))
(x2)*(sqrt(5-x2))
x2*sqrt5-x2
(x²)*(sqrt(5-x²))
(x to the power of 2)*(sqrt(5-x to the power of 2))
(x^2)(sqrt(5-x^2))
(x2)(sqrt(5-x2))
x2sqrt5-x2
x^2sqrt5-x^2
Similar expressions
(x^2)*(sqrt(5+x^2))

The derivative of the function/ (x^2)*(sqrt(5-x^2))

Derivative of (x^2)*(sqrt(5-x^2))

The solution

You have entered [src]

      ________
 2   /      2 
x *\/  5 - x

$$x^{2} \sqrt{5 - x^{2}}$$

  /      ________\
d | 2   /      2 |
--\x *\/  5 - x  /
dx

$$\frac{d}{d x} x^{2} \sqrt{5 - x^{2}}$$

Transform

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = x^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{2}$ goes to $2 x$
$g{\left(x \right)} = \sqrt{5 - x^{2}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 5 - x^{2}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 - x^{2}\right)$ :
  1. Differentiate $5 - x^{2}$ term by term:
    1. The derivative of the constant $5$ 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:
  
  $- \frac{x}{\sqrt{5 - x^{2}}}$
The result is: $- \frac{x^{3}}{\sqrt{5 - x^{2}}} + 2 x \sqrt{5 - x^{2}}$
Now simplify:

$\frac{x \left(10 - 3 x^{2}\right)}{\sqrt{5 - x^{2}}}$

The answer is:

$\frac{x \left(10 - 3 x^{2}\right)}{\sqrt{5 - x^{2}}}$

The graph

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Plot the graph f'(x)

The first derivative [src]

        3              ________
       x              /      2 
- ----------- + 2*x*\/  5 - x  
     ________                  
    /      2                   
  \/  5 - x

$$- \frac{x^{3}}{\sqrt{5 - x^{2}}} + 2 x \sqrt{5 - x^{2}}$$

Simplify

The second derivative [src]

                                 /         2  \
                               2 |        x   |
                              x *|-1 + -------|
     ________          2         |           2|
    /      2        4*x          \     -5 + x /
2*\/  5 - x   - ----------- + -----------------
                   ________         ________   
                  /      2         /      2    
                \/  5 - x        \/  5 - x

$$\frac{x^{2} \left(\frac{x^{2}}{x^{2} - 5} - 1\right)}{\sqrt{5 - x^{2}}} - \frac{4 x^{2}}{\sqrt{5 - x^{2}}} + 2 \sqrt{5 - x^{2}}$$

Simplify

The third derivative [src]

    /                  /         2  \\
    |                2 |        x   ||
    |               x *|-1 + -------||
    |          2       |           2||
    |       2*x        \     -5 + x /|
3*x*|-4 + ------- + -----------------|
    |           2              2     |
    \     -5 + x          5 - x      /
--------------------------------------
                ________              
               /      2               
             \/  5 - x

$$\frac{3 x \left(\frac{2 x^{2}}{x^{2} - 5} + \frac{x^{2} \left(\frac{x^{2}}{x^{2} - 5} - 1\right)}{5 - x^{2}} - 4\right)}{\sqrt{5 - x^{2}}}$$

Simplify

The graph

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Derivative of (x^2)*(sqrt(5-x^2))

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Piecewise:

The solution