Derivative of y=sinx+cosx*sinx-cosx

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

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

d                                  
--(sin(x) + cos(x)*sin(x) - cos(x))
dx

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

Transform

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

-sin(x) - 4*cos(x)*sin(x) + cos(x)

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

Simplify

The third derivative [src]

                        2           2   
-cos(x) - sin(x) - 4*cos (x) + 4*sin (x)

$$4 \sin^{2}{\left(x \right)} - 4 \cos^{2}{\left(x \right)} - \sin{\left(x \right)} - \cos{\left(x \right)}$$

Simplify

The graph