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The derivative of the function/ cosln(1-x^2)

Derivative of cosln(1-x^2)

The solution

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   /   /     2\\
cos\log\1 - x //

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

Transform

Detail solution

Let $u = \log{\left(1 - x^{2} \right)}$ .
The derivative of cosine is negative sine:

$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(1 - x^{2} \right)}$ :
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 of the chain rule is:

$\frac{2 x \sin{\left(\log{\left(1 - x^{2} \right)} \right)}}{1 - x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /                        2    /   /     2\\      2    /   /     2\\\
  |     /   /     2\\   2*x *cos\log\1 - x //   2*x *sin\log\1 - x //|
2*|- sin\log\1 - x // - --------------------- + ---------------------|
  |                                  2                       2       |
  \                            -1 + x                  -1 + x        /
----------------------------------------------------------------------
                                     2                                
                               -1 + x

$$\frac{2 \left(\frac{2 x^{2} \sin{\left(\log{\left(1 - x^{2} \right)} \right)}}{x^{2} - 1} - \frac{2 x^{2} \cos{\left(\log{\left(1 - x^{2} \right)} \right)}}{x^{2} - 1} - \sin{\left(\log{\left(1 - x^{2} \right)} \right)}\right)}{x^{2} - 1}$$

Simplify

The third derivative [src]

    /                                               2    /   /     2\\      2    /   /     2\\\
    |       /   /     2\\        /   /     2\\   2*x *sin\log\1 - x //   6*x *cos\log\1 - x //|
4*x*|- 3*cos\log\1 - x // + 3*sin\log\1 - x // - --------------------- + ---------------------|
    |                                                         2                       2       |
    \                                                   -1 + x                  -1 + x        /
-----------------------------------------------------------------------------------------------
                                                    2                                          
                                           /      2\                                           
                                           \-1 + x /

$$\frac{4 x \left(- \frac{2 x^{2} \sin{\left(\log{\left(1 - x^{2} \right)} \right)}}{x^{2} - 1} + \frac{6 x^{2} \cos{\left(\log{\left(1 - x^{2} \right)} \right)}}{x^{2} - 1} + 3 \sin{\left(\log{\left(1 - x^{2} \right)} \right)} - 3 \cos{\left(\log{\left(1 - x^{2} \right)} \right)}\right)}{\left(x^{2} - 1\right)^{2}}$$

Simplify

The graph

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Derivative of cosln(1-x^2)

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The solution