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Identical expressions
sqrt(cosx+3x^ four)
square root of ( co sinus of e of x plus 3x to the power of 4)
square root of ( co sinus of e of x plus 3x to the power of four)
√(cosx+3x^4)
sqrt(cosx+3x4)
sqrtcosx+3x4
sqrt(cosx+3x⁴)
sqrtcosx+3x^4
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sqrt(cosx-3x^4)

The derivative of the function/ sqrt(cosx+3x^4)

Derivative of sqrt(cosx+3x^4)

The solution

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   _______________
  /             4 
\/  cos(x) + 3*x

$$\sqrt{3 x^{4} + \cos{\left(x \right)}}$$

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Detail solution

Let $u = 3 x^{4} + \cos{\left(x \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x^{4} + \cos{\left(x \right)}\right)$ :
1. Differentiate $3 x^{4} + \cos{\left(x \right)}$ term by term:
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
    So, the result is: $12 x^{3}$
  The result is: $12 x^{3} - \sin{\left(x \right)}$
The result of the chain rule is:

$\frac{12 x^{3} - \sin{\left(x \right)}}{2 \sqrt{3 x^{4} + \cos{\left(x \right)}}}$
Now simplify:

$\frac{12 x^{3} - \sin{\left(x \right)}}{2 \sqrt{3 x^{4} + \cos{\left(x \right)}}}$

The answer is:

$\frac{12 x^{3} - \sin{\left(x \right)}}{2 \sqrt{3 x^{4} + \cos{\left(x \right)}}}$

The graph

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The first derivative [src]

     3   sin(x)   
  6*x  - ------   
           2      
------------------
   _______________
  /             4 
\/  cos(x) + 3*x

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

Simplify

The second derivative [src]

                                  2
                 /              3\ 
    2   cos(x)   \-sin(x) + 12*x / 
18*x  - ------ - ------------------
          2        /   4         \ 
                 4*\3*x  + cos(x)/ 
-----------------------------------
            _______________        
           /    4                  
         \/  3*x  + cos(x)

$$\frac{18 x^{2} - \frac{\left(12 x^{3} - \sin{\left(x \right)}\right)^{2}}{4 \left(3 x^{4} + \cos{\left(x \right)}\right)} - \frac{\cos{\left(x \right)}}{2}}{\sqrt{3 x^{4} + \cos{\left(x \right)}}}$$

Simplify

The third derivative [src]

                                   3                                        
                  /              3\      /              2\ /              3\
sin(x)          3*\-sin(x) + 12*x /    3*\-cos(x) + 36*x /*\-sin(x) + 12*x /
------ + 36*x + -------------------- - -------------------------------------
  2                               2                /   4         \          
                   /   4         \               4*\3*x  + cos(x)/          
                 8*\3*x  + cos(x)/                                          
----------------------------------------------------------------------------
                                _______________                             
                               /    4                                       
                             \/  3*x  + cos(x)

$$\frac{36 x - \frac{3 \left(36 x^{2} - \cos{\left(x \right)}\right) \left(12 x^{3} - \sin{\left(x \right)}\right)}{4 \left(3 x^{4} + \cos{\left(x \right)}\right)} + \frac{3 \left(12 x^{3} - \sin{\left(x \right)}\right)^{3}}{8 \left(3 x^{4} + \cos{\left(x \right)}\right)^{2}} + \frac{\sin{\left(x \right)}}{2}}{\sqrt{3 x^{4} + \cos{\left(x \right)}}}$$

Simplify

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Derivative of sqrt(cosx+3x^4)

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The solution