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The derivative of the function/ sqrt(1-8sinx/8)

Derivative of sqrt(1-8sinx/8)

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

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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} 
sqrt(1 - 8*sin(x)/8)
Detail solution

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

  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(−8sin⁡(x)8+1)\frac{d}{d x} \left(- \frac{8 \sin{\left(x \right)}}{8} + 1\right):

    1. Differentiate −8sin⁡(x)8+1- \frac{8 \sin{\left(x \right)}}{8} + 1 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. 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)}

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

  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)1−sin⁡(x)41−sin⁡(x)\frac{2 \sin{\left(x \right)} - \frac{\cos^{2}{\left(x \right)}}{1 - \sin{\left(x \right)}}}{4 \sqrt{1 - \sin{\left(x \right)}}}
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)1−sin⁡(x)−3cos⁡2(x)(1−sin⁡(x))2)cos⁡(x)81−sin⁡(x)\frac{\left(4 + \frac{6 \sin{\left(x \right)}}{1 - \sin{\left(x \right)}} - \frac{3 \cos^{2}{\left(x \right)}}{\left(1 - \sin{\left(x \right)}\right)^{2}}\right) \cos{\left(x \right)}}{8 \sqrt{1 - \sin{\left(x \right)}}}
Simplify
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