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The derivative of the function/ sin(x)*cos(x)+x

Derivative of sin(x)*cos(x)+x

The solution

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sin(x)*cos(x) + x

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

sin(x)*cos(x) + x

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of sin(x)*cos(x)+x