Derivative of lnsin(2^x)

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   /   / x\\
log\sin\2 //

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

log(sin(2^x))

Detail solution

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

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

$\frac{2^{x} \log{\left(2 \right)}}{\tan{\left(2^{x} \right)}}$

The answer is:

$\frac{2^{x} \log{\left(2 \right)}}{\tan{\left(2^{x} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$\frac{2^{x} \log{\left(2 \right)} \cos{\left(2^{x} \right)}}{\sin{\left(2^{x} \right)}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Identical expressions

Derivative of lnsin(2^x)

The graph:

Piecewise:

The solution