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Derivative of:
Derivative of cos(x/3)
Derivative of √x
Derivative of xcos(x)
Derivative of 50/x
Identical expressions
three *cos(x)+3x*sin(x)
3 multiply by co sinus of e of (x) plus 3x multiply by sinus of (x)
three multiply by co sinus of e of (x) plus 3x multiply by sinus of (x)
3cos(x)+3xsin(x)
3cosx+3xsinx
Similar expressions
3*cos(x)-3x*sin(x)
3*cosx+3x*sinx

The derivative of the function/ 3*cos(x)+3x*sin(x)

Derivative of 3cos(x)+3xsin(x)

The solution

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

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

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

Detail solution

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

The answer is:

$3 x \cos{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

3*x*cos(x)

$$3 x \cos{\left(x \right)}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify