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Derivative of:
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Identical expressions
(two *(cost+(t*(sint))))
(2 multiply by ( co sinus of e of t plus (t multiply by ( sinus of t))))
(two multiply by ( co sinus of e of t plus (t multiply by ( sinus of t))))
(2(cost+(t(sint))))
2cost+tsint
Similar expressions
(2*(cost-(t*(sint))))

The derivative of the function/ (2*(cost+(t*(sint))))

Derivative of (2(cost+(t(sint))))

The solution

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

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

2*(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: $2 t \cos{\left(t \right)}$

The answer is:

$2 t \cos{\left(t \right)}$

The graph

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The first derivative [src]

2*t*cos(t)

$$2 t \cos{\left(t \right)}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify