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Derivative of ln(x+3)^3
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Derivative of -cos(2*x)
Identical expressions
sinln((2x- three)/(x+ four))
sinus of ln((2x minus 3) divide by (x plus 4))
sinus of ln((2x minus three) divide by (x plus four))
sinln2x-3/x+4
sinln((2x-3) divide by (x+4))
Similar expressions
sinln((2x+3)/(x+4))
sinln((2x-3)/(x-4))

The derivative of the function/ sinln((2x-3)/(x+4))

Derivative of sinln((2x-3)/(x+4))

The solution

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   /   /2*x - 3\\
sin|log|-------||
   \   \ x + 4 //

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

sin(log((2*x - 3)/(x + 4)))

Detail solution

Let $u = \log{\left(\frac{2 x - 3}{x + 4} \right)}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(\frac{2 x - 3}{x + 4} \right)}$ :
1. Let $u = \frac{2 x - 3}{x + 4}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{2 x - 3}{x + 4}$ :
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = 2 x - 3$ and $g{\left(x \right)} = x + 4$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Differentiate $2 x - 3$ term by term:
      1. The derivative of the constant $-3$ is zero.
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
      The result is: $2$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Differentiate $x + 4$ term by term:
      1. The derivative of the constant $4$ is zero.
      2. Apply the power rule: $x$ goes to $1$
      The result is: $1$
    Now plug in to the quotient rule:
    
    $\frac{11}{\left(x + 4\right)^{2}}$
  The result of the chain rule is:
  
  $\frac{11 \left(x + 4\right)}{\left(x + 4\right)^{2} \left(2 x - 3\right)}$
The result of the chain rule is:

$\frac{11 \left(x + 4\right) \cos{\left(\log{\left(\frac{2 x - 3}{x + 4} \right)} \right)}}{\left(x + 4\right)^{2} \left(2 x - 3\right)}$
Now simplify:

$\frac{11 \cos{\left(\log{\left(\frac{2 x - 3}{x + 4} \right)} \right)}}{\left(x + 4\right) \left(2 x - 3\right)}$

The answer is:

$\frac{11 \cos{\left(\log{\left(\frac{2 x - 3}{x + 4} \right)} \right)}}{\left(x + 4\right) \left(2 x - 3\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        /  2     2*x - 3 \    /   /2*x - 3\\
(x + 4)*|----- - --------|*cos|log|-------||
        |x + 4          2|    \   \ x + 4 //
        \        (x + 4) /                  
--------------------------------------------
                  2*x - 3

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

Simplify

The second derivative [src]

               /     /   /-3 + 2*x\\        /   /-3 + 2*x\\   /    -3 + 2*x\    /   /-3 + 2*x\\\
               |  cos|log|--------||   2*cos|log|--------||   |2 - --------|*sin|log|--------|||
/    -3 + 2*x\ |     \   \ 4 + x  //        \   \ 4 + x  //   \     4 + x  /    \   \ 4 + x  //|
|2 - --------|*|- ------------------ - -------------------- - ---------------------------------|
\     4 + x  / \        4 + x                -3 + 2*x                      -3 + 2*x            /
------------------------------------------------------------------------------------------------
                                            -3 + 2*x

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

Simplify

The third derivative [src]

               /                                                            2                                                                                                                      \
               |     /   /-3 + 2*x\\        /   /-3 + 2*x\\   /    -3 + 2*x\     /   /-3 + 2*x\\        /   /-3 + 2*x\\     /    -3 + 2*x\    /   /-3 + 2*x\\     /    -3 + 2*x\    /   /-3 + 2*x\\|
               |2*cos|log|--------||   8*cos|log|--------||   |2 - --------| *cos|log|--------||   4*cos|log|--------||   6*|2 - --------|*sin|log|--------||   3*|2 - --------|*sin|log|--------|||
/    -3 + 2*x\ |     \   \ 4 + x  //        \   \ 4 + x  //   \     4 + x  /     \   \ 4 + x  //        \   \ 4 + x  //     \     4 + x  /    \   \ 4 + x  //     \     4 + x  /    \   \ 4 + x  //|
|2 - --------|*|-------------------- + -------------------- - ---------------------------------- + -------------------- + ----------------------------------- + -----------------------------------|
\     4 + x  / |             2                       2                             2                (-3 + 2*x)*(4 + x)                          2                        (-3 + 2*x)*(4 + x)        |
               \      (4 + x)              (-3 + 2*x)                    (-3 + 2*x)                                                   (-3 + 2*x)                                                   /
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                              -3 + 2*x

$$\frac{\left(2 - \frac{2 x - 3}{x + 4}\right) \left(- \frac{\left(2 - \frac{2 x - 3}{x + 4}\right)^{2} \cos{\left(\log{\left(\frac{2 x - 3}{x + 4} \right)} \right)}}{\left(2 x - 3\right)^{2}} + \frac{6 \left(2 - \frac{2 x - 3}{x + 4}\right) \sin{\left(\log{\left(\frac{2 x - 3}{x + 4} \right)} \right)}}{\left(2 x - 3\right)^{2}} + \frac{3 \left(2 - \frac{2 x - 3}{x + 4}\right) \sin{\left(\log{\left(\frac{2 x - 3}{x + 4} \right)} \right)}}{\left(x + 4\right) \left(2 x - 3\right)} + \frac{8 \cos{\left(\log{\left(\frac{2 x - 3}{x + 4} \right)} \right)}}{\left(2 x - 3\right)^{2}} + \frac{4 \cos{\left(\log{\left(\frac{2 x - 3}{x + 4} \right)} \right)}}{\left(x + 4\right) \left(2 x - 3\right)} + \frac{2 \cos{\left(\log{\left(\frac{2 x - 3}{x + 4} \right)} \right)}}{\left(x + 4\right)^{2}}\right)}{2 x - 3}$$

Simplify

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Derivative of sinln((2x-3)/(x+4))

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The solution