Derivative of ln(sin4x)*sin4x

The solution

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log(sin(4*x))*sin(4*x)

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

log(sin(4*x))*sin(4*x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \log{\left(\sin{\left(4 x \right)} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(4 x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
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:
  
  $\frac{4 \cos{\left(4 x \right)}}{\sin{\left(4 x \right)}}$
$g{\left(x \right)} = \sin{\left(4 x \right)}$ ; 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:
  
  $4 \cos{\left(4 x \right)}$
The result is: $4 \log{\left(\sin{\left(4 x \right)} \right)} \cos{\left(4 x \right)} + 4 \cos{\left(4 x \right)}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

4*cos(4*x) + 4*cos(4*x)*log(sin(4*x))

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

    /       2                     \         
    |    cos (4*x)                |         
-64*|4 + --------- + log(sin(4*x))|*cos(4*x)
    |       2                     |         
    \    sin (4*x)                /

$$- 64 \left(\log{\left(\sin{\left(4 x \right)} \right)} + 4 + \frac{\cos^{2}{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)}}\right) \cos{\left(4 x \right)}$$

Simplify

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Derivative of ln(sin4x)*sin4x

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The solution