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Derivative of ln((x^2+a)^0.5)
Identical expressions
ln((x^ two +a)^ zero . five)
ln((x squared plus a) to the power of 0.5)
ln((x to the power of two plus a) to the power of zero . five)
ln((x2+a)0.5)
lnx2+a0.5
ln((x²+a)^0.5)
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lnx^2+a^0.5
Similar expressions
ln((x^2-a)^0.5)

The derivative of the function/ ln((x^2+a)^0.5)

Derivative of ln((x^2+a)^0.5)

The solution

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   /   ________\
   |  /  2     |
log\\/  x  + a /

$$\log{\left(\sqrt{a + x^{2}} \right)}$$

log(sqrt(x^2 + a))

Detail solution

Let $u = \sqrt{a + x^{2}}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} \sqrt{a + x^{2}}$ :
1. Let $u = a + x^{2}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} \left(a + x^{2}\right)$ :
  1. Differentiate $a + x^{2}$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    2. The derivative of the constant $a$ is zero.
    The result is: $2 x$
  The result of the chain rule is:
  
  $\frac{x}{\sqrt{a + x^{2}}}$
The result of the chain rule is:

$\frac{x}{a + x^{2}}$
Now simplify:

$\frac{x}{a + x^{2}}$

The answer is:

$\frac{x}{a + x^{2}}$

The first derivative [src]

  x   
------
 2    
x  + a

$$\frac{x}{a + x^{2}}$$

Simplify

The second derivative [src]

        2 
     2*x  
1 - ------
         2
    a + x 
----------
       2  
  a + x

$$\frac{- \frac{2 x^{2}}{a + x^{2}} + 1}{a + x^{2}}$$

Simplify

The third derivative [src]

    /         2 \
    |      4*x  |
2*x*|-3 + ------|
    |          2|
    \     a + x /
-----------------
            2    
    /     2\     
    \a + x /

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

Simplify

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Derivative of ln((x^2+a)^0.5)

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