Derivative of ln(sin(2x+4))

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log(sin(2*x + 4))

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

log(sin(2*x + 4))

Detail solution

Let $u = \sin{\left(2 x + 4 \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(2 x + 4 \right)}$ :
1. Let $u = 2 x + 4$ .
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 $\frac{d}{d x} \left(2 x + 4\right)$ :
  1. Differentiate $2 x + 4$ term by term:
    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: $2$
    2. The derivative of the constant $4$ is zero.
    The result is: $2$
  The result of the chain rule is:
  
  $2 \cos{\left(2 x + 4 \right)}$
The result of the chain rule is:

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

$\frac{2}{\tan{\left(2 x + 4 \right)}}$

The answer is:

$\frac{2}{\tan{\left(2 x + 4 \right)}}$

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of ln(sin(2x+4))

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The solution