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Derivative of:
Derivative of y=log((sin^2)(4x))
Derivative of 8*cos(x)+(30/pi)*x+19
Derivative of x/sqrt(x^2-1)
Derivative of 6x^3
Identical expressions
y=log((sin^ two)(4x))
y equally logarithm of (( sinus of squared )(4x))
y equally logarithm of (( sinus of to the power of two)(4x))
y=log((sin2)(4x))
y=logsin24x
y=log((sin²)(4x))
y=log((sin to the power of 2)(4x))
y=logsin^24x

The derivative of the function/ y=log((sin^2)(4x))

Derivative of y=log((sin^2)(4x))

The solution

You have entered [src]

   /   2       \
log\sin (x)*4*x/

\log{\left(\sin^{2}{\left(x \right)} 4 x \right)}

d /   /   2       \\
--\log\sin (x)*4*x//
dx

\frac{d}{d x} \log{\left(\sin^{2}{\left(x \right)} 4 x \right)}

Transform

Detail solution

Let $u = \sin^{2}{\left(x \right)} 4 x$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin^{2}{\left(x \right)} 4 x$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the power rule: $x$ goes to $1$
    $g{\left(x \right)} = \sin^{2}{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = \sin{\left(x \right)}$ .
    2. Apply the power rule: $u^{2}$ goes to $2 u$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
      1. The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      The result of the chain rule is:
      
      $2 \sin{\left(x \right)} \cos{\left(x \right)}$
    The result is: $2 x \sin{\left(x \right)} \cos{\left(x \right)} + \sin^{2}{\left(x \right)}$
  So, the result is: $8 x \sin{\left(x \right)} \cos{\left(x \right)} + 4 \sin^{2}{\left(x \right)}$
The result of the chain rule is:

$\frac{8 x \sin{\left(x \right)} \cos{\left(x \right)} + 4 \sin^{2}{\left(x \right)}}{4 x \sin^{2}{\left(x \right)}}$
Now simplify:

$\frac{2}{\tan{\left(x \right)}} + \frac{1}{x}$

The answer is:

\frac{2}{\tan{\left(x \right)}} + \frac{1}{x}

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     2                       
4*sin (x) + 8*x*cos(x)*sin(x)
-----------------------------
                2            
         4*x*sin (x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                             2           2                                         /     2           2                     \            /     2           2                     \        2                                                                   \
  |2*x*cos(x) + sin(x)   - 3*cos (x) + 3*sin (x) + 4*x*cos(x)*sin(x)                4*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/*cos(x)   2*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/   3*cos (x)*(2*x*cos(x) + sin(x))   2*(2*x*cos(x) + sin(x))*cos(x)         |
2*|------------------- - ------------------------------------------- + 2*x*cos(x) - -------------------------------------------------- - ------------------------------------------- + ------------------------------- + ------------------------------ + sin(x)|
  |          2                              sin(x)                                                          2                                              x*sin(x)                                   2                             x*sin(x)                    |
  \         x                                                                                            sin (x)                                                                                   sin (x)                                                      /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                             x*sin(x)

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

Simplify

The graph

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Identical expressions

Derivative of y=log((sin^2)(4x))

The graph:

Piecewise:

The solution