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

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

The solution

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x*cos(x) + sin(x)

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

x*cos(x) + sin(x)

Detail solution

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

$$- (x \cos{\left(x \right)} + 3 \sin{\left(x \right)})$$

Simplify

The third derivative [src]

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

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

Simplify

The graph

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