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Derivative of:
Derivative of x^12
Derivative of -e^x
Derivative of e^(2-x)
Derivative of x^2-4*x
Identical expressions
lnsin(x)- one /4cosx
ln sinus of (x) minus 1 divide by 4 co sinus of e of x
ln sinus of (x) minus one divide by 4 co sinus of e of x
lnsinx-1/4cosx
lnsin(x)-1 divide by 4cosx
Similar expressions
ln(sin(x-1/4*cos(x)))
lnsin(x)+1/4cosx
ln((sin(x)-1/4cos(x)))
lnsinx-1/4cosx

The derivative of the function/ lnsin(x)-1/4cosx

Derivative of lnsin(x)-1/4cosx

The solution

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              cos(x)
log(sin(x)) - ------
                4

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

log(sin(x)) - cos(x)/4

Detail solution

Differentiate $\log{\left(\sin{\left(x \right)} \right)} - \frac{\cos{\left(x \right)}}{4}$ term by term:
1. Let $u = \sin{\left(x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
  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:
  
  $\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  So, the result is: $\frac{\sin{\left(x \right)}}{4}$
The result is: $\frac{\sin{\left(x \right)}}{4} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
Now simplify:

$\frac{\sin{\left(x \right)}}{4} + \frac{1}{\tan{\left(x \right)}}$

The answer is:

$\frac{\sin{\left(x \right)}}{4} + \frac{1}{\tan{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                 2   
     cos(x)   cos (x)
-1 + ------ - -------
       4         2   
              sin (x)

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

Simplify

The third derivative [src]

                3              
  sin(x)   2*cos (x)   2*cos(x)
- ------ + --------- + --------
    4          3        sin(x) 
            sin (x)

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

Simplify

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Derivative of lnsin(x)-1/4cosx

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The solution