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Derivative of:
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Identical expressions
sqrt(x)*ln(one -x^ two)
square root of (x) multiply by ln(1 minus x squared )
square root of (x) multiply by ln(one minus x to the power of two)
√(x)*ln(1-x^2)
sqrt(x)*ln(1-x2)
sqrtx*ln1-x2
sqrt(x)*ln(1-x²)
sqrt(x)*ln(1-x to the power of 2)
sqrt(x)ln(1-x^2)
sqrt(x)ln(1-x2)
sqrtxln1-x2
sqrtxln1-x^2
Similar expressions
sqrt(x)*ln(1+x^2)

The derivative of the function/ sqrt(x)*ln(1-x^2)

Derivative of sqrt(x)*ln(1-x^2)

The solution

You have entered [src]

  ___    /     2\
\/ x *log\1 - x /

$$\sqrt{x} \log{\left(1 - x^{2} \right)}$$

sqrt(x)*log(1 - x^2)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sqrt{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
$g{\left(x \right)} = \log{\left(1 - x^{2} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 1 - x^{2}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. 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:
  
  $- \frac{2 x}{1 - x^{2}}$
The result is: $- \frac{2 x^{\frac{3}{2}}}{1 - x^{2}} + \frac{\log{\left(1 - x^{2} \right)}}{2 \sqrt{x}}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

   /     2\      3/2
log\1 - x /   2*x   
----------- - ------
      ___          2
  2*\/ x      1 - x

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

Simplify

The second derivative [src]

                                /          2 \
                            ___ |       2*x  |
                        2*\/ x *|-1 + -------|
    ___      /     2\           |           2|
2*\/ x    log\1 - x /           \     -1 + x /
------- - ----------- - ----------------------
      2         3/2                  2        
-1 + x       4*x               -1 + x

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

Simplify

The third derivative [src]

                                        /          2 \          /          2 \
                                        |       2*x  |      3/2 |       4*x  |
                                      3*|-1 + -------|   4*x   *|-3 + -------|
                           /     2\     |           2|          |           2|
          3           3*log\1 - x /     \     -1 + x /          \     -1 + x /
- ----------------- + ------------- - ---------------- + ---------------------
      ___ /      2\          5/2        ___ /      2\                   2     
  2*\/ x *\-1 + x /       8*x         \/ x *\-1 + x /          /      2\      
                                                               \-1 + x /

$$\frac{4 x^{\frac{3}{2}} \left(\frac{4 x^{2}}{x^{2} - 1} - 3\right)}{\left(x^{2} - 1\right)^{2}} - \frac{3 \left(\frac{2 x^{2}}{x^{2} - 1} - 1\right)}{\sqrt{x} \left(x^{2} - 1\right)} - \frac{3}{2 \sqrt{x} \left(x^{2} - 1\right)} + \frac{3 \log{\left(1 - x^{2} \right)}}{8 x^{\frac{5}{2}}}$$

Simplify

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Derivative of sqrt(x)*ln(1-x^2)

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The solution