Mister Exam

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The derivative of the function/ 8*sint*cost

Derivative of 8sintcost

The solution

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8*sin(t)*cos(t)

$$8 \sin{\left(t \right)} \cos{\left(t \right)}$$

(8*sin(t))*cos(t)

Detail solution

Apply the product rule:

$\frac{d}{d t} f{\left(t \right)} g{\left(t \right)} = f{\left(t \right)} \frac{d}{d t} g{\left(t \right)} + g{\left(t \right)} \frac{d}{d t} f{\left(t \right)}$

$f{\left(t \right)} = 8 \sin{\left(t \right)}$ ; to find $\frac{d}{d t} f{\left(t \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of sine is cosine:
    
    $\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}$
  So, the result is: $8 \cos{\left(t \right)}$
$g{\left(t \right)} = \cos{\left(t \right)}$ ; 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: $- 8 \sin^{2}{\left(t \right)} + 8 \cos^{2}{\left(t \right)}$
Now simplify:

$8 \cos{\left(2 t \right)}$

The answer is:

$8 \cos{\left(2 t \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2           2   
- 8*sin (t) + 8*cos (t)

$$- 8 \sin^{2}{\left(t \right)} + 8 \cos^{2}{\left(t \right)}$$

Simplify

The second derivative [src]

-32*cos(t)*sin(t)

$$- 32 \sin{\left(t \right)} \cos{\left(t \right)}$$

Simplify

The third derivative [src]

   /   2         2   \
32*\sin (t) - cos (t)/

$$32 \left(\sin^{2}{\left(t \right)} - \cos^{2}{\left(t \right)}\right)$$

Simplify