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Similar expressions
4(sin(t)+t*cos(t))

The derivative of the function/ 4(sin(t)-t*cos(t))

Derivative of 4(sin(t)-t*cos(t))

The solution

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4*(sin(t) - t*cos(t))

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

4*(sin(t) - t*cos(t))

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Differentiate $- t \cos{\left(t \right)} + \sin{\left(t \right)}$ term by term:
  1. The derivative of sine is cosine:
    
    $\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. 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 $1$
      $g{\left(t \right)} = \cos{\left(t \right)}$ ; to find $\frac{d}{d t} g{\left(t \right)}$ :
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}$
      The result is: $- t \sin{\left(t \right)} + \cos{\left(t \right)}$
    So, the result is: $t \sin{\left(t \right)} - \cos{\left(t \right)}$
  The result is: $t \sin{\left(t \right)}$
So, the result is: $4 t \sin{\left(t \right)}$

The answer is:

$4 t \sin{\left(t \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

4*t*sin(t)

$$4 t \sin{\left(t \right)}$$

Simplify

The second derivative [src]

4*(t*cos(t) + sin(t))

$$4 \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right)$$

Simplify

The third derivative [src]

4*(2*cos(t) - t*sin(t))

$$4 \left(- t \sin{\left(t \right)} + 2 \cos{\left(t \right)}\right)$$

Simplify