Mister Exam

Lang:

The derivative of the function/ 4sinxcosx

Derivative of 4sinxcosx

The solution

You have entered [src]

4*sin(x)*cos(x)

$$4 \sin{\left(x \right)} \cos{\left(x \right)}$$

(4*sin(x))*cos(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 4 \sin{\left(x \right)}$ ; 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. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  So, the result is: $4 \cos{\left(x \right)}$
$g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
The result is: $- 4 \sin^{2}{\left(x \right)} + 4 \cos^{2}{\left(x \right)}$
Now simplify:

$4 \cos{\left(2 x \right)}$

The answer is:

$4 \cos{\left(2 x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2           2   
- 4*sin (x) + 4*cos (x)

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

Simplify

The second derivative [src]

-16*cos(x)*sin(x)

$$- 16 \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplify

The third derivative [src]

   /   2         2   \
16*\sin (x) - cos (x)/

$$16 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)$$

Simplify