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Derivative of (1-cos4x)/sin4x
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Identical expressions
(one -cos4x)/sin4x
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(one minus co sinus of e of 4x) divide by sinus of 4x
1-cos4x/sin4x
(1-cos4x) divide by sin4x
Similar expressions
(1+cos4x)/sin4x

The derivative of the function/ (1-cos4x)/sin4x

Derivative of (1-cos4x)/sin4x

The solution

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1 - cos(4*x)
------------
  sin(4*x)

$$\frac{- \cos{\left(4 x \right)} + 1}{\sin{\left(4 x \right)}}$$

d /1 - cos(4*x)\
--|------------|
dx\  sin(4*x)  /

$$\frac{d}{d x} \frac{- \cos{\left(4 x \right)} + 1}{\sin{\left(4 x \right)}}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 1 - \cos{\left(4 x \right)}$ and $g{\left(x \right)} = \sin{\left(4 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $1 - \cos{\left(4 x \right)}$ 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 $u = 4 x$ .
    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 $\frac{d}{d x} 4 x$ :
      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: $4$
      The result of the chain rule is:
      
      $- 4 \sin{\left(4 x \right)}$
    So, the result is: $4 \sin{\left(4 x \right)}$
  The result is: $4 \sin{\left(4 x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 4 x$ .
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 $\frac{d}{d x} 4 x$ :
  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: $4$
  The result of the chain rule is:
  
  $4 \cos{\left(4 x \right)}$
Now plug in to the quotient rule:

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

$\frac{4}{\cos{\left(4 x \right)} + 1}$

The answer is:

$\frac{4}{\cos{\left(4 x \right)} + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

    //         2     \                           \
    ||    2*cos (4*x)|                           |
-16*||1 + -----------|*(-1 + cos(4*x)) + cos(4*x)|
    ||        2      |                           |
    \\     sin (4*x) /                           /
--------------------------------------------------
                     sin(4*x)

$$- \frac{16 \left(\left(1 + \frac{2 \cos^{2}{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)}}\right) \left(\cos{\left(4 x \right)} - 1\right) + \cos{\left(4 x \right)}\right)}{\sin{\left(4 x \right)}}$$

Simplify

The third derivative [src]

   /                                  /         2     \         \
   |                                  |    6*cos (4*x)|         |
   |                  (-1 + cos(4*x))*|5 + -----------|*cos(4*x)|
   |         2                        |        2      |         |
   |    3*cos (4*x)                   \     sin (4*x) /         |
64*|2 + ----------- + ------------------------------------------|
   |        2                            2                      |
   \     sin (4*x)                    sin (4*x)                 /

$$64 \left(\frac{\left(5 + \frac{6 \cos^{2}{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)}}\right) \left(\cos{\left(4 x \right)} - 1\right) \cos{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)}} + 2 + \frac{3 \cos^{2}{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)}}\right)$$

Simplify

The graph

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Derivative of (1-cos4x)/sin4x

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The solution