Derivative of y=(sin4x)^arctg1/x

The solution

You have entered [src]

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

$$\frac{\sin^{\operatorname{atan}{\left(1 \right)}}{\left(4 x \right)}}{x}$$

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

$$\frac{d}{d x} \frac{\sin^{\operatorname{atan}{\left(1 \right)}}{\left(4 x \right)}}{x}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin^{\operatorname{atan}{\left(1 \right)}}{\left(4 x \right)}$ and $g{\left(x \right)} = x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(4 x \right)}$ .
2. Apply the power rule: $u^{\operatorname{atan}{\left(1 \right)}}$ goes to $\frac{u^{\operatorname{atan}{\left(1 \right)}} \operatorname{atan}{\left(1 \right)}}{u}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(4 x \right)}$ :
  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:
    
    $4 \cos{\left(4 x \right)}$
  The result of the chain rule is:
  
  $\frac{4 \sin^{\operatorname{atan}{\left(1 \right)}}{\left(4 x \right)} \cos{\left(4 x \right)} \operatorname{atan}{\left(1 \right)}}{\sin{\left(4 x \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                  /       /       2           2             \                             \
     atan(1)      |1      |    cos (4*x)   cos (4*x)*atan(1)|           4*atan(1)*cos(4*x)|
2*sin       (4*x)*|-- - 8*|1 + --------- - -----------------|*atan(1) - ------------------|
                  | 2     |       2               2         |               x*sin(4*x)    |
                  \x      \    sin (4*x)       sin (4*x)    /                             /
-------------------------------------------------------------------------------------------
                                             x

$$\frac{2 \left(- 8 \cdot \left(1 - \frac{\cos^{2}{\left(4 x \right)} \operatorname{atan}{\left(1 \right)}}{\sin^{2}{\left(4 x \right)}} + \frac{\cos^{2}{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)}}\right) \operatorname{atan}{\left(1 \right)} - \frac{4 \cos{\left(4 x \right)} \operatorname{atan}{\left(1 \right)}}{x \sin{\left(4 x \right)}} + \frac{1}{x^{2}}\right) \sin^{\operatorname{atan}{\left(1 \right)}}{\left(4 x \right)}}{x}$$

Simplify

The third derivative [src]

                  /          /       2           2             \                                    /                     2            2       2             2             \                 \
                  |          |    cos (4*x)   cos (4*x)*atan(1)|                                    |                2*cos (4*x)   atan (1)*cos (4*x)   3*cos (4*x)*atan(1)|                 |
                  |       24*|1 + --------- - -----------------|*atan(1)                         32*|2 - 3*atan(1) + ----------- + ------------------ - -------------------|*atan(1)*cos(4*x)|
                  |          |       2               2         |                                    |                    2                2                     2          |                 |
     atan(1)      |  3       \    sin (4*x)       sin (4*x)    /           12*atan(1)*cos(4*x)      \                 sin (4*x)        sin (4*x)             sin (4*x)     /                 |
2*sin       (4*x)*|- -- + ---------------------------------------------- + ------------------- + --------------------------------------------------------------------------------------------|
                  |   3                         x                               2                                                          sin(4*x)                                          |
                  \  x                                                         x *sin(4*x)                                                                                                   /
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                              x

$$\frac{2 \cdot \left(\frac{32 \left(- 3 \operatorname{atan}{\left(1 \right)} + 2 - \frac{3 \cos^{2}{\left(4 x \right)} \operatorname{atan}{\left(1 \right)}}{\sin^{2}{\left(4 x \right)}} + \frac{\cos^{2}{\left(4 x \right)} \operatorname{atan}^{2}{\left(1 \right)}}{\sin^{2}{\left(4 x \right)}} + \frac{2 \cos^{2}{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)}}\right) \cos{\left(4 x \right)} \operatorname{atan}{\left(1 \right)}}{\sin{\left(4 x \right)}} + \frac{24 \cdot \left(1 - \frac{\cos^{2}{\left(4 x \right)} \operatorname{atan}{\left(1 \right)}}{\sin^{2}{\left(4 x \right)}} + \frac{\cos^{2}{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)}}\right) \operatorname{atan}{\left(1 \right)}}{x} + \frac{12 \cos{\left(4 x \right)} \operatorname{atan}{\left(1 \right)}}{x^{2} \sin{\left(4 x \right)}} - \frac{3}{x^{3}}\right) \sin^{\operatorname{atan}{\left(1 \right)}}{\left(4 x \right)}}{x}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Derivative of y=(sin4x)^arctg1/x

The graph:

Piecewise:

The solution