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

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

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
    ______________
   /     8*sin(x) 
  /  1 - -------- 
\/          8
−8sin⁡(x)8+1\sqrt{- \frac{8 \sin{\left(x \right)}}{8} + 1} 
  /    ______________\
d |   /     8*sin(x) |
--|  /  1 - -------- |
dx\\/          8     /
ddx−8sin⁡(x)8+1\frac{d}{d x} \sqrt{- \frac{8 \sin{\left(x \right)}}{8} + 1}
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Detail solution

  1. Let u=1−8sin⁡(x)8u = 1 - \frac{8 \sin{\left(x \right)}}{8}.

  2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

  3. Then, apply the chain rule. Multiply by ddx(1−8sin⁡(x)8)\frac{d}{d x} \left(1 - \frac{8 \sin{\left(x \right)}}{8}\right):

    1. Differentiate 1−8sin⁡(x)81 - \frac{8 \sin{\left(x \right)}}{8} term by term:

      1. The derivative of the constant 11 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:

          ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=8sin⁡(x)f{\left(x \right)} = 8 \sin{\left(x \right)} and g(x)=8g{\left(x \right)} = 8.

          To find ddxf(x)\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.

            1. The derivative of sine is cosine:

              ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

            So, the result is: 8cos⁡(x)8 \cos{\left(x \right)}

          To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

          1. The derivative of the constant 88 is zero.

          Now plug in to the quotient rule:

          cos⁡(x)\cos{\left(x \right)}

        So, the result is: −cos⁡(x)- \cos{\left(x \right)}

      The result is: −cos⁡(x)- \cos{\left(x \right)}

    The result of the chain rule is:

    −cos⁡(x)21−8sin⁡(x)8- \frac{\cos{\left(x \right)}}{2 \sqrt{1 - \frac{8 \sin{\left(x \right)}}{8}}}

  4. Now simplify:

    −cos⁡(x)21−sin⁡(x)- \frac{\cos{\left(x \right)}}{2 \sqrt{1 - \sin{\left(x \right)}}}

The answer is:

−cos⁡(x)21−sin⁡(x)- \frac{\cos{\left(x \right)}}{2 \sqrt{1 - \sin{\left(x \right)}}}

The graph
02468-8-6-4-2-10102.5-2.5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
      -cos(x)       
--------------------
      ______________
     /     8*sin(x) 
2*  /  1 - -------- 
  \/          8
−cos⁡(x)2−8sin⁡(x)8+1- \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)
2sin⁡(x)−cos⁡2(x)−sin⁡(x)+14−sin⁡(x)+1\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)
(4+6sin⁡(x)−sin⁡(x)+1−3cos⁡2(x)(−sin⁡(x)+1)2)cos⁡(x)8−sin⁡(x)+1\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
Derivative of sqrt(1-8*sin(x)/8)
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