Derivative of sqrt(5-4sinx)

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  ______________
\/ 5 - 4*sin(x)

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

d /  ______________\
--\\/ 5 - 4*sin(x) /
dx

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

Transform

Detail solution

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

$- \frac{2 \cos{\left(x \right)}}{\sqrt{5 - 4 \sin{\left(x \right)}}}$
Now simplify:

$- \frac{2 \cos{\left(x \right)}}{\sqrt{5 - 4 \sin{\left(x \right)}}}$

The answer is:

$- \frac{2 \cos{\left(x \right)}}{\sqrt{5 - 4 \sin{\left(x \right)}}}$

The graph

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The first derivative [src]

   -2*cos(x)    
----------------
  ______________
\/ 5 - 4*sin(x)

$$- \frac{2 \cos{\left(x \right)}}{\sqrt{- 4 \sin{\left(x \right)} + 5}}$$

Simplify

The second derivative [src]

  /        2              \
  |   2*cos (x)           |
2*|- ------------ + sin(x)|
  \  5 - 4*sin(x)         /
---------------------------
        ______________     
      \/ 5 - 4*sin(x)

$$\frac{2 \left(\sin{\left(x \right)} - \frac{2 \cos^{2}{\left(x \right)}}{- 4 \sin{\left(x \right)} + 5}\right)}{\sqrt{- 4 \sin{\left(x \right)} + 5}}$$

Simplify

The third derivative [src]

  /             2                    \       
  |       12*cos (x)       6*sin(x)  |       
2*|1 - --------------- + ------------|*cos(x)
  |                  2   5 - 4*sin(x)|       
  \    (5 - 4*sin(x))                /       
---------------------------------------------
                 ______________              
               \/ 5 - 4*sin(x)

$$\frac{2 \cdot \left(1 + \frac{6 \sin{\left(x \right)}}{- 4 \sin{\left(x \right)} + 5} - \frac{12 \cos^{2}{\left(x \right)}}{\left(- 4 \sin{\left(x \right)} + 5\right)^{2}}\right) \cos{\left(x \right)}}{\sqrt{- 4 \sin{\left(x \right)} + 5}}$$

Simplify

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Derivative of sqrt(5-4sinx)

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