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The derivative of the function/ sinx(1+cosx)

Derivative of sinx(1+cosx)

The solution

You have entered [src]

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

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

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

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

-(1 + 4*cos(x))*sin(x)

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

Simplify

The third derivative [src]

       2           2                         
- 3*cos (x) + 4*sin (x) - (1 + cos(x))*cos(x)

$$- \left(\cos{\left(x \right)} + 1\right) \cos{\left(x \right)} + 4 \sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}$$

Simplify

The graph

Derivative of sinx(1+cosx)