Derivative of tg(arcsin(x))

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tan(asin(x))

$$\tan{\left(\operatorname{asin}{\left(x \right)} \right)}$$

tan(asin(x))

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(\operatorname{asin}{\left(x \right)} \right)} = \frac{x}{\sqrt{1 - x^{2}}}$
Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x$ and $g{\left(x \right)} = \sqrt{1 - x^{2}}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 1 - x^{2}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - x^{2}\right)$ :
  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:
  
  $- \frac{x}{\sqrt{1 - x^{2}}}$
Now plug in to the quotient rule:

$\frac{\frac{x^{2}}{\sqrt{1 - x^{2}}} + \sqrt{1 - x^{2}}}{1 - x^{2}}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

       2         
1 + tan (asin(x))
-----------------
      ________   
     /      2    
   \/  1 - x

$$\frac{\tan^{2}{\left(\operatorname{asin}{\left(x \right)} \right)} + 1}{\sqrt{1 - x^{2}}}$$

Simplify

The second derivative [src]

/       2         \ /     x        2*tan(asin(x))\
\1 + tan (asin(x))/*|----------- - --------------|
                    |        3/2            2    |
                    |/     2\         -1 + x     |
                    \\1 - x /                    /

$$\left(\frac{x}{\left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{2 \tan{\left(\operatorname{asin}{\left(x \right)} \right)}}{x^{2} - 1}\right) \left(\tan^{2}{\left(\operatorname{asin}{\left(x \right)} \right)} + 1\right)$$

Simplify

The third derivative [src]

                    /                /       2         \          2           2                            \
/       2         \ |     1        2*\1 + tan (asin(x))/       3*x       4*tan (asin(x))   6*x*tan(asin(x))|
\1 + tan (asin(x))/*|----------- + --------------------- + ----------- + --------------- + ----------------|
                    |        3/2                3/2                5/2             3/2                 2   |
                    |/     2\           /     2\           /     2\        /     2\           /      2\    |
                    \\1 - x /           \1 - x /           \1 - x /        \1 - x /           \-1 + x /    /

$$\left(\tan^{2}{\left(\operatorname{asin}{\left(x \right)} \right)} + 1\right) \left(\frac{3 x^{2}}{\left(1 - x^{2}\right)^{\frac{5}{2}}} + \frac{6 x \tan{\left(\operatorname{asin}{\left(x \right)} \right)}}{\left(x^{2} - 1\right)^{2}} + \frac{2 \left(\tan^{2}{\left(\operatorname{asin}{\left(x \right)} \right)} + 1\right)}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{4 \tan^{2}{\left(\operatorname{asin}{\left(x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{1}{\left(1 - x^{2}\right)^{\frac{3}{2}}}\right)$$

Simplify

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Derivative of tg(arcsin(x))

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The solution