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Derivative of:
Derivative of -4*y-y^2
Derivative of 4-2x
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Derivative of e^sin(x)^(2)
Identical expressions
sqrt(one - eight *sin(x)/ eight)
square root of (1 minus 8 multiply by sinus of (x) divide by 8)
square root of (one minus eight multiply by sinus of (x) divide by eight)
√(1-8*sin(x)/8)
sqrt(1-8sin(x)/8)
sqrt1-8sinx/8
sqrt(1-8*sin(x) divide by 8)
Similar expressions
sqrt(1+8*sin(x)/8)
sqrt(1-8sinx/8)
sqrt(1-8*sinx/8)

The derivative of the function/ sqrt(1-8*sin(x)/8)

Derivative of sqrt(1-8*sin(x)/8)

The solution

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    ______________
   /     8*sin(x) 
  /  1 - -------- 
\/          8

$$\sqrt{- \frac{8 \sin{\left(x \right)}}{8} + 1}$$

  /    ______________\
d |   /     8*sin(x) |
--|  /  1 - -------- |
dx\\/          8     /

$$\frac{d}{d x} \sqrt{- \frac{8 \sin{\left(x \right)}}{8} + 1}$$

Transform

Detail solution

Let $u = 1 - \frac{8 \sin{\left(x \right)}}{8}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - \frac{8 \sin{\left(x \right)}}{8}\right)$ :
1. Differentiate $1 - \frac{8 \sin{\left(x \right)}}{8}$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = 8 \sin{\left(x \right)}$ and $g{\left(x \right)} = 8$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
        
        So, the result is: $8 \cos{\left(x \right)}$
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      1. The derivative of the constant $8$ is zero.
      Now plug in to the quotient rule:
      
      $\cos{\left(x \right)}$
    So, the result is: $- \cos{\left(x \right)}$
  The result is: $- \cos{\left(x \right)}$
The result of the chain rule is:

$- \frac{\cos{\left(x \right)}}{2 \sqrt{1 - \frac{8 \sin{\left(x \right)}}{8}}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      -cos(x)       
--------------------
      ______________
     /     8*sin(x) 
2*  /  1 - -------- 
  \/          8

$$- \frac{\cos{\left(x \right)}}{2 \sqrt{- \frac{8 \sin{\left(x \right)}}{8} + 1}}$$

Simplify

The second derivative [src]

               2     
            cos (x)  
2*sin(x) - ----------
           1 - sin(x)
---------------------
       ____________  
   4*\/ 1 - sin(x)

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

Simplify

The third derivative [src]

/           2                  \       
|      3*cos (x)      6*sin(x) |       
|4 - ------------- + ----------|*cos(x)
|                2   1 - sin(x)|       
\    (1 - sin(x))              /       
---------------------------------------
                ____________           
            8*\/ 1 - sin(x)

$$\frac{\left(4 + \frac{6 \sin{\left(x \right)}}{- \sin{\left(x \right)} + 1} - \frac{3 \cos^{2}{\left(x \right)}}{\left(- \sin{\left(x \right)} + 1\right)^{2}}\right) \cos{\left(x \right)}}{8 \sqrt{- \sin{\left(x \right)} + 1}}$$

Simplify

The graph

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Derivative of sqrt(1-8*sin(x)/8)

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The solution