Derivative of sinx+4(√(1-sinx))-5

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             ____________    
sin(x) + 4*\/ 1 - sin(x)  - 5

$$\left(4 \sqrt{1 - \sin{\left(x \right)}} + \sin{\left(x \right)}\right) - 5$$

sin(x) + 4*sqrt(1 - sin(x)) - 5

Detail solution

Differentiate $\left(4 \sqrt{1 - \sin{\left(x \right)}} + \sin{\left(x \right)}\right) - 5$ term by term:
1. Differentiate $4 \sqrt{1 - \sin{\left(x \right)}} + \sin{\left(x \right)}$ term by term:
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = 1 - \sin{\left(x \right)}$ .
    2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - \sin{\left(x \right)}\right)$ :
      1. Differentiate $1 - \sin{\left(x \right)}$ term by term:
        
        The derivative of the constant $1$ is zero.
        
        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: $- \cos{\left(x \right)}$
        
        The result is: $- \cos{\left(x \right)}$
      The result of the chain rule is:
      
      $- \frac{\cos{\left(x \right)}}{2 \sqrt{1 - \sin{\left(x \right)}}}$
    So, the result is: $- \frac{2 \cos{\left(x \right)}}{\sqrt{1 - \sin{\left(x \right)}}}$
  The result is: $\cos{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{\sqrt{1 - \sin{\left(x \right)}}}$
2. The derivative of the constant $-5$ is zero.
The result is: $\cos{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{\sqrt{1 - \sin{\left(x \right)}}}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

/                                                 2       \       
|           2              3*sin(x)          3*cos (x)    |       
|-1 + -------------- + --------------- - -----------------|*cos(x)
|       ____________               3/2                 5/2|       
\     \/ 1 - sin(x)    (1 - sin(x))      2*(1 - sin(x))   /

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

Simplify

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Derivative of sinx+4(√(1-sinx))-5

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