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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   /   /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
  1. Let .

  2. The derivative of sine is cosine:

  3. Then, apply the chain rule. Multiply by :

    1. Let .

    2. The derivative of is .

    3. Then, apply the chain rule. Multiply by :

      1. Apply the quotient rule, which is:

        and .

        To find :

        1. Differentiate term by term:

          1. The derivative of the constant is zero.

          2. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: goes to

            So, the result is:

          The result is:

        To find :

        1. Differentiate term by term:

          1. The derivative of the constant is zero.

          2. Apply the power rule: goes to

          The result is:

        Now plug in to the quotient rule:

      The result of the chain rule is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

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