Derivative of (1-2sin2x)\(cos6x)

The solution

You have entered [src]

1 - 2*sin(2*x)
--------------
   cos(6*x)

$$\frac{1 - 2 \sin{\left(2 x \right)}}{\cos{\left(6 x \right)}}$$

(1 - 2*sin(2*x))/cos(6*x)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 1 - 2 \sin{\left(2 x \right)}$ and $g{\left(x \right)} = \cos{\left(6 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $1 - 2 \sin{\left(2 x \right)}$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = 2 x$ .
    2. The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
      The result of the chain rule is:
      
      $2 \cos{\left(2 x \right)}$
    So, the result is: $- 4 \cos{\left(2 x \right)}$
  The result is: $- 4 \cos{\left(2 x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 6 x$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 6 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $6$
  The result of the chain rule is:
  
  $- 6 \sin{\left(6 x \right)}$
Now plug in to the quotient rule:

$\frac{6 \left(1 - 2 \sin{\left(2 x \right)}\right) \sin{\left(6 x \right)} - 4 \cos{\left(2 x \right)} \cos{\left(6 x \right)}}{\cos^{2}{\left(6 x \right)}}$
Now simplify:

$\frac{2 \left(3 \sin{\left(6 x \right)} - 4 \cos{\left(4 x \right)} + 2 \cos{\left(8 x \right)}\right)}{\cos^{2}{\left(6 x \right)}}$

The answer is:

$\frac{2 \left(3 \sin{\left(6 x \right)} - 4 \cos{\left(4 x \right)} + 2 \cos{\left(8 x \right)}\right)}{\cos^{2}{\left(6 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  4*cos(2*x)   6*(1 - 2*sin(2*x))*sin(6*x)
- ---------- + ---------------------------
   cos(6*x)                2              
                        cos (6*x)

$$\frac{6 \left(1 - 2 \sin{\left(2 x \right)}\right) \sin{\left(6 x \right)}}{\cos^{2}{\left(6 x \right)}} - \frac{4 \cos{\left(2 x \right)}}{\cos{\left(6 x \right)}}$$

Simplify

The second derivative [src]

  /               /         2     \                                         \
  |               |    2*sin (6*x)|                     12*cos(2*x)*sin(6*x)|
4*|2*sin(2*x) - 9*|1 + -----------|*(-1 + 2*sin(2*x)) - --------------------|
  |               |        2      |                           cos(6*x)      |
  \               \     cos (6*x) /                                         /
-----------------------------------------------------------------------------
                                   cos(6*x)

$$\frac{4 \left(- 9 \left(\frac{2 \sin^{2}{\left(6 x \right)}}{\cos^{2}{\left(6 x \right)}} + 1\right) \left(2 \sin{\left(2 x \right)} - 1\right) + 2 \sin{\left(2 x \right)} - \frac{12 \sin{\left(6 x \right)} \cos{\left(2 x \right)}}{\cos{\left(6 x \right)}}\right)}{\cos{\left(6 x \right)}}$$

Simplify

The third derivative [src]

  /                                                                                         /         2     \         \
  |                                                                                         |    6*sin (6*x)|         |
  |                                                                    27*(-1 + 2*sin(2*x))*|5 + -----------|*sin(6*x)|
  |                /         2     \                                                        |        2      |         |
  |                |    2*sin (6*x)|            18*sin(2*x)*sin(6*x)                        \     cos (6*x) /         |
8*|2*cos(2*x) - 54*|1 + -----------|*cos(2*x) + -------------------- - -----------------------------------------------|
  |                |        2      |                  cos(6*x)                             cos(6*x)                   |
  \                \     cos (6*x) /                                                                                  /
-----------------------------------------------------------------------------------------------------------------------
                                                        cos(6*x)

$$\frac{8 \left(- 54 \left(\frac{2 \sin^{2}{\left(6 x \right)}}{\cos^{2}{\left(6 x \right)}} + 1\right) \cos{\left(2 x \right)} - \frac{27 \left(\frac{6 \sin^{2}{\left(6 x \right)}}{\cos^{2}{\left(6 x \right)}} + 5\right) \left(2 \sin{\left(2 x \right)} - 1\right) \sin{\left(6 x \right)}}{\cos{\left(6 x \right)}} + \frac{18 \sin{\left(2 x \right)} \sin{\left(6 x \right)}}{\cos{\left(6 x \right)}} + 2 \cos{\left(2 x \right)}\right)}{\cos{\left(6 x \right)}}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of (1-2sin2x)\(cos6x)

The graph:

Piecewise:

The solution