Derivative of lnsin(2x+5)

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log(sin(2*x + 5))

$$\log{\left(\sin{\left(2 x + 5 \right)} \right)}$$

d                    
--(log(sin(2*x + 5)))
dx

$$\frac{d}{d x} \log{\left(\sin{\left(2 x + 5 \right)} \right)}$$

Transform

Detail solution

Let $u = \sin{\left(2 x + 5 \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(2 x + 5 \right)}$ :
1. Let $u = 2 x + 5$ .
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 $\frac{d}{d x} \left(2 x + 5\right)$ :
  1. Differentiate $2 x + 5$ 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 $5$ is zero.
    The result is: $2$
  The result of the chain rule is:
  
  $2 \cos{\left(2 x + 5 \right)}$
The result of the chain rule is:

$\frac{2 \cos{\left(2 x + 5 \right)}}{\sin{\left(2 x + 5 \right)}}$
Now simplify:

$\frac{2}{\tan{\left(2 x + 5 \right)}}$

The answer is:

$\frac{2}{\tan{\left(2 x + 5 \right)}}$

The graph

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Plot the graph f'(x)

The first derivative [src]

2*cos(2*x + 5)
--------------
 sin(2*x + 5)

$$\frac{2 \cos{\left(2 x + 5 \right)}}{\sin{\left(2 x + 5 \right)}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /       2         \             
   |    cos (5 + 2*x)|             
16*|1 + -------------|*cos(5 + 2*x)
   |       2         |             
   \    sin (5 + 2*x)/             
-----------------------------------
            sin(5 + 2*x)

$$\frac{16 \cdot \left(1 + \frac{\cos^{2}{\left(2 x + 5 \right)}}{\sin^{2}{\left(2 x + 5 \right)}}\right) \cos{\left(2 x + 5 \right)}}{\sin{\left(2 x + 5 \right)}}$$

Simplify

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Derivative of lnsin(2x+5)

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The solution