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Identical expressions
two ^cos((3x+ one)^ zero . five)
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two to the power of co sinus of e of ((3x plus one) to the power of zero . five)
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Similar expressions
2^cos((3x-1)^0.5)

The derivative of the function/ 2^cos((3x+1)^0.5)

Derivative of 2^cos((3x+1)^0.5)

The solution

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    /  _________\
 cos\\/ 3*x + 1 /
2

$$2^{\cos{\left(\sqrt{3 x + 1} \right)}}$$

2^cos(sqrt(3*x + 1))

Detail solution

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

$- \frac{3 \cdot 2^{\cos{\left(\sqrt{3 x + 1} \right)}} \log{\left(2 \right)} \sin{\left(\sqrt{3 x + 1} \right)}}{2 \sqrt{3 x + 1}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       /  _________\                        
    cos\\/ 3*x + 1 /           /  _________\
-3*2                *log(2)*sin\\/ 3*x + 1 /
--------------------------------------------
                   _________                
               2*\/ 3*x + 1

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

Simplify

The second derivative [src]

      /  _________\ /   /  _________\      /  _________\      2/  _________\       \       
   cos\\/ 1 + 3*x / |sin\\/ 1 + 3*x /   cos\\/ 1 + 3*x /   sin \\/ 1 + 3*x /*log(2)|       
9*2                *|---------------- - ---------------- + ------------------------|*log(2)
                    |           3/2         1 + 3*x                1 + 3*x         |       
                    \  (1 + 3*x)                                                   /       
-------------------------------------------------------------------------------------------
                                             4

$$\frac{9 \cdot 2^{\cos{\left(\sqrt{3 x + 1} \right)}} \left(\frac{\log{\left(2 \right)} \sin^{2}{\left(\sqrt{3 x + 1} \right)}}{3 x + 1} - \frac{\cos{\left(\sqrt{3 x + 1} \right)}}{3 x + 1} + \frac{\sin{\left(\sqrt{3 x + 1} \right)}}{\left(3 x + 1\right)^{\frac{3}{2}}}\right) \log{\left(2 \right)}}{4}$$

Simplify

The third derivative [src]

       /  _________\ /   /  _________\        /  _________\        /  _________\      2       3/  _________\        2/  _________\               /  _________\           /  _________\\       
    cos\\/ 1 + 3*x / |sin\\/ 1 + 3*x /   3*sin\\/ 1 + 3*x /   3*cos\\/ 1 + 3*x /   log (2)*sin \\/ 1 + 3*x /   3*sin \\/ 1 + 3*x /*log(2)   3*cos\\/ 1 + 3*x /*log(2)*sin\\/ 1 + 3*x /|       
27*2                *|---------------- - ------------------ + ------------------ - ------------------------- - -------------------------- + ------------------------------------------|*log(2)
                     |           3/2                 5/2                   2                       3/2                          2                                   3/2               |       
                     \  (1 + 3*x)           (1 + 3*x)             (1 + 3*x)               (1 + 3*x)                    (1 + 3*x)                           (1 + 3*x)                  /       
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                              8

$$\frac{27 \cdot 2^{\cos{\left(\sqrt{3 x + 1} \right)}} \left(- \frac{3 \log{\left(2 \right)} \sin^{2}{\left(\sqrt{3 x + 1} \right)}}{\left(3 x + 1\right)^{2}} + \frac{3 \cos{\left(\sqrt{3 x + 1} \right)}}{\left(3 x + 1\right)^{2}} - \frac{\log{\left(2 \right)}^{2} \sin^{3}{\left(\sqrt{3 x + 1} \right)}}{\left(3 x + 1\right)^{\frac{3}{2}}} + \frac{3 \log{\left(2 \right)} \sin{\left(\sqrt{3 x + 1} \right)} \cos{\left(\sqrt{3 x + 1} \right)}}{\left(3 x + 1\right)^{\frac{3}{2}}} + \frac{\sin{\left(\sqrt{3 x + 1} \right)}}{\left(3 x + 1\right)^{\frac{3}{2}}} - \frac{3 \sin{\left(\sqrt{3 x + 1} \right)}}{\left(3 x + 1\right)^{\frac{5}{2}}}\right) \log{\left(2 \right)}}{8}$$

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Derivative of 2^cos((3x+1)^0.5)

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The solution