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The derivative of the function/ √1+sinx(1-sinx)

Derivative of √1+sinx(1-sinx)

The solution

You have entered [src]

  ___                      
\/ 1  + sin(x)*(1 - sin(x))

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

d /  ___                      \
--\\/ 1  + sin(x)*(1 - sin(x))/
dx

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

Transform

Detail solution

Differentiate $\left(1 - \sin{\left(x \right)}\right) \sin{\left(x \right)} + \sqrt{1}$ term by term:
1. The derivative of the constant $\sqrt{1}$ is zero.
2. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  $g{\left(x \right)} = 1 - \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $1 - \sin{\left(x \right)}$ 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. 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 is: $\left(1 - \sin{\left(x \right)}\right) \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}$
The result is: $\left(1 - \sin{\left(x \right)}\right) \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}$
Now simplify:

$- \sin{\left(2 x \right)} + \cos{\left(x \right)}$

The answer is:

$- \sin{\left(2 x \right)} + \cos{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

(1 - sin(x))*cos(x) - cos(x)*sin(x)

$$\left(- \sin{\left(x \right)} + 1\right) \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplify

The second derivative [src]

   2           2                          
sin (x) - 2*cos (x) + (-1 + sin(x))*sin(x)

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

Simplify

The third derivative [src]

(-1 + 8*sin(x))*cos(x)

$$\left(8 \sin{\left(x \right)} - 1\right) \cos{\left(x \right)}$$

Simplify

The graph

Derivative of √1+sinx(1-sinx)