Derivative of sqrt(1+sin(14x))

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  _______________
\/ 1 + sin(14*x)

$$\sqrt{\sin{\left(14 x \right)} + 1}$$

sqrt(1 + sin(14*x))

Detail solution

Let $u = \sin{\left(14 x \right)} + 1$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\sin{\left(14 x \right)} + 1\right)$ :
1. Differentiate $\sin{\left(14 x \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Let $u = 14 x$ .
  3. The derivative of sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} 14 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $14$
    The result of the chain rule is:
    
    $14 \cos{\left(14 x \right)}$
  The result is: $14 \cos{\left(14 x \right)}$
The result of the chain rule is:

$\frac{7 \cos{\left(14 x \right)}}{\sqrt{\sin{\left(14 x \right)} + 1}}$

The answer is:

$\frac{7 \cos{\left(14 x \right)}}{\sqrt{\sin{\left(14 x \right)} + 1}}$

The graph

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The first derivative [src]

   7*cos(14*x)   
-----------------
  _______________
\/ 1 + sin(14*x)

$$\frac{7 \cos{\left(14 x \right)}}{\sqrt{\sin{\left(14 x \right)} + 1}}$$

Simplify

The second derivative [src]

    /                   2       \
    |                cos (14*x) |
-49*|2*sin(14*x) + -------------|
    \              1 + sin(14*x)/
---------------------------------
          _______________        
        \/ 1 + sin(14*x)

$$- \frac{49 \left(2 \sin{\left(14 x \right)} + \frac{\cos^{2}{\left(14 x \right)}}{\sin{\left(14 x \right)} + 1}\right)}{\sqrt{\sin{\left(14 x \right)} + 1}}$$

Simplify

The third derivative [src]

    /            2                        \          
    |       3*cos (14*x)      6*sin(14*x) |          
343*|-4 + ---------------- + -------------|*cos(14*x)
    |                    2   1 + sin(14*x)|          
    \     (1 + sin(14*x))                 /          
-----------------------------------------------------
                    _______________                  
                  \/ 1 + sin(14*x)

$$\frac{343 \left(-4 + \frac{6 \sin{\left(14 x \right)}}{\sin{\left(14 x \right)} + 1} + \frac{3 \cos^{2}{\left(14 x \right)}}{\left(\sin{\left(14 x \right)} + 1\right)^{2}}\right) \cos{\left(14 x \right)}}{\sqrt{\sin{\left(14 x \right)} + 1}}$$

Simplify

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Derivative of sqrt(1+sin(14x))

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