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Similar expressions
3*(cos(t)-(t)*sin(t))

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

Derivative of 3(cos(t)+(t)sin(t))

The solution

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3*(cos(t) + t*sin(t))

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

3*(cos(t) + t*sin(t))

Detail solution

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

3*t*cos(t)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify