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Identical expressions
root^ four (two x^2+ six)^ five
root to the power of 4(2x squared plus 6) to the power of 5
root to the power of four (two x squared plus six) to the power of five
root4(2x2+6)5
root42x2+65
root⁴(2x²+6)⁵
root to the power of 4(2x to the power of 2+6) to the power of 5
root^42x^2+6^5
Similar expressions
root^4(2x^2-6)^5

The derivative of the function/ root^4(2x^2+6)^5

Derivative of root^4(2x^2+6)^5

The solution

You have entered [src]

             1024
   __________    
  /    2         
\/  2*x  + 6

$$\left(\sqrt{2 x^{2} + 6}\right)^{1024}$$

Transform

Detail solution

Let $u = \sqrt{2 x^{2} + 6}$ .
Apply the power rule: $u^{1024}$ goes to $1024 u^{1023}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{2 x^{2} + 6}$ :
1. Let $u = 2 x^{2} + 6$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x^{2} + 6\right)$ :
  1. Differentiate $2 x^{2} + 6$ term by term:
    1. 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: $4 x$
    2. The derivative of the constant $6$ is zero.
    The result is: $4 x$
  The result of the chain rule is:
  
  $\frac{2 x}{\sqrt{2 x^{2} + 6}}$
The result of the chain rule is:

$2048 x \left(2 x^{2} + 6\right)^{511}$
Now simplify:

$13729595320261219429963801598162786434538870600286610818788926918371086366795312104245119281322909109954592622782961716074243975999433287625148056582230114304 x \left(x^{2} + 3\right)^{511}$

The answer is:

$13729595320261219429963801598162786434538870600286610818788926918371086366795312104245119281322909109954592622782961716074243975999433287625148056582230114304 x \left(x^{2} + 3\right)^{511}$

The first derivative [src]

                 512
       /   2    \   
2048*x*\2*x  + 6/   
--------------------
         2          
      2*x  + 6

$$\frac{2048 x \left(2 x^{2} + 6\right)^{512}}{2 x^{2} + 6}$$

Simplify

The second derivative [src]

                                                                                                                                                                       510              
                                                                                                                                                               /     2\    /          2\
13729595320261219429963801598162786434538870600286610818788926918371086366795312104245119281322909109954592622782961716074243975999433287625148056582230114304*\3 + x /   *\3 + 1023*x /

$$13729595320261219429963801598162786434538870600286610818788926918371086366795312104245119281322909109954592622782961716074243975999433287625148056582230114304 \left(x^{2} + 3\right)^{510} \left(1023 x^{2} + 3\right)$$

Simplify

The third derivative [src]

                                                                                                                                                                            509             
                                                                                                                                                                    /     2\    /         2\
42094939251920898772269015699967103208296177260478748770406849931725750800594426911615535716536039331120780981452560621483632030414262459858703941481117530456064*x*\3 + x /   *\3 + 341*x /

$$42094939251920898772269015699967103208296177260478748770406849931725750800594426911615535716536039331120780981452560621483632030414262459858703941481117530456064 x \left(x^{2} + 3\right)^{509} \left(341 x^{2} + 3\right)$$

Simplify

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Derivative of root^4(2x^2+6)^5

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