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sinln(x^ three + one)
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Similar expressions
sinln(x^3-1)

The derivative of the function/ sinln(x^3+1)

Derivative of sinln(x^3+1)

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

$$\frac{d}{d x} \sin{\left(\log{\left(x^{3} + 1 \right)} \right)}$$

Transform

Detail solution

Let $u = \log{\left(x^{3} + 1 \right)}$ .
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 $\frac{d}{d x} \log{\left(x^{3} + 1 \right)}$ :
1. Let $u = x^{3} + 1$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{3} + 1\right)$ :
  1. Differentiate $x^{3} + 1$ term by term:
    1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
    2. The derivative of the constant $1$ is zero.
    The result is: $3 x^{2}$
  The result of the chain rule is:
  
  $\frac{3 x^{2}}{x^{3} + 1}$
The result of the chain rule is:

$\frac{3 x^{2} \cos{\left(\log{\left(x^{3} + 1 \right)} \right)}}{x^{3} + 1}$
Now simplify:

$\frac{3 x^{2} \cos{\left(\log{\left(x^{3} + 1 \right)} \right)}}{x^{3} + 1}$

The answer is:

$\frac{3 x^{2} \cos{\left(\log{\left(x^{3} + 1 \right)} \right)}}{x^{3} + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2    /   / 3    \\
3*x *cos\log\x  + 1//
---------------------
         3           
        x  + 1

$$\frac{3 x^{2} \cos{\left(\log{\left(x^{3} + 1 \right)} \right)}}{x^{3} + 1}$$

Simplify

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

$$\frac{3 x \left(- \frac{3 x^{3} \sin{\left(\log{\left(x^{3} + 1 \right)} \right)}}{x^{3} + 1} - \frac{3 x^{3} \cos{\left(\log{\left(x^{3} + 1 \right)} \right)}}{x^{3} + 1} + 2 \cos{\left(\log{\left(x^{3} + 1 \right)} \right)}\right)}{x^{3} + 1}$$

Simplify

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

$$\frac{3 \cdot \left(\frac{27 x^{6} \sin{\left(\log{\left(x^{3} + 1 \right)} \right)}}{\left(x^{3} + 1\right)^{2}} + \frac{9 x^{6} \cos{\left(\log{\left(x^{3} + 1 \right)} \right)}}{\left(x^{3} + 1\right)^{2}} - \frac{18 x^{3} \sin{\left(\log{\left(x^{3} + 1 \right)} \right)}}{x^{3} + 1} - \frac{18 x^{3} \cos{\left(\log{\left(x^{3} + 1 \right)} \right)}}{x^{3} + 1} + 2 \cos{\left(\log{\left(x^{3} + 1 \right)} \right)}\right)}{x^{3} + 1}$$

Simplify

The graph

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Derivative of sinln(x^3+1)

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Piecewise:

The solution