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sinln(x^3+1)

Derivative of sinln(x^3+1)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   /   / 3    \\
sin\log\x  + 1//
$$\sin{\left(\log{\left(x^{3} + 1 \right)} \right)}$$
d /   /   / 3    \\\
--\sin\log\x  + 1///
dx                  
$$\frac{d}{d x} \sin{\left(\log{\left(x^{3} + 1 \right)} \right)}$$
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. Differentiate term by term:

        1. Apply the power rule: goes to

        2. The derivative of the constant is zero.

        The result is:

      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    /   / 3    \\
3*x *cos\log\x  + 1//
---------------------
         3           
        x  + 1       
$$\frac{3 x^{2} \cos{\left(\log{\left(x^{3} + 1 \right)} \right)}}{x^{3} + 1}$$
The second derivative [src]
    /                        3    /   /     3\\      3    /   /     3\\\
    |     /   /     3\\   3*x *cos\log\1 + x //   3*x *sin\log\1 + x //|
3*x*|2*cos\log\1 + x // - --------------------- - ---------------------|
    |                                  3                       3       |
    \                             1 + x                   1 + x        /
------------------------------------------------------------------------
                                      3                                 
                                 1 + x                                  
$$\frac{3 x \left(- \frac{3 x^{3} \sin{\left(\log{\left(x^{3} + 1 \right)} \right)}}{x^{3} + 1} - \frac{3 x^{3} \cos{\left(\log{\left(x^{3} + 1 \right)} \right)}}{x^{3} + 1} + 2 \cos{\left(\log{\left(x^{3} + 1 \right)} \right)}\right)}{x^{3} + 1}$$
The third derivative [src]
  /                         3    /   /     3\\       3    /   /     3\\      6    /   /     3\\       6    /   /     3\\\
  |     /   /     3\\   18*x *cos\log\1 + x //   18*x *sin\log\1 + x //   9*x *cos\log\1 + x //   27*x *sin\log\1 + x //|
3*|2*cos\log\1 + x // - ---------------------- - ---------------------- + --------------------- + ----------------------|
  |                                  3                        3                         2                       2       |
  |                             1 + x                    1 + x                  /     3\                /     3\        |
  \                                                                             \1 + x /                \1 + x /        /
-------------------------------------------------------------------------------------------------------------------------
                                                               3                                                         
                                                          1 + x                                                          
$$\frac{3 \cdot \left(\frac{27 x^{6} \sin{\left(\log{\left(x^{3} + 1 \right)} \right)}}{\left(x^{3} + 1\right)^{2}} + \frac{9 x^{6} \cos{\left(\log{\left(x^{3} + 1 \right)} \right)}}{\left(x^{3} + 1\right)^{2}} - \frac{18 x^{3} \sin{\left(\log{\left(x^{3} + 1 \right)} \right)}}{x^{3} + 1} - \frac{18 x^{3} \cos{\left(\log{\left(x^{3} + 1 \right)} \right)}}{x^{3} + 1} + 2 \cos{\left(\log{\left(x^{3} + 1 \right)} \right)}\right)}{x^{3} + 1}$$
The graph
Derivative of sinln(x^3+1)