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y=sin(x)*log2x^3

Derivative of y=sin(x)*log2x^3

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
          3     
sin(x)*log (2*x)
$$\log{\left(2 x \right)}^{3} \sin{\left(x \right)}$$
sin(x)*log(2*x)^3
Detail solution
  1. Apply the product rule:

    ; to find :

    1. The derivative of sine is cosine:

    ; to find :

    1. Let .

    2. Apply the power rule: goes to

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

      1. Let .

      2. The derivative of is .

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

        1. 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 of the chain rule is:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
                        2            
   3               3*log (2*x)*sin(x)
log (2*x)*cos(x) + ------------------
                           x         
$$\log{\left(2 x \right)}^{3} \cos{\left(x \right)} + \frac{3 \log{\left(2 x \right)}^{2} \sin{\left(x \right)}}{x}$$
The second derivative [src]
/     2               3*(-2 + log(2*x))*sin(x)   6*cos(x)*log(2*x)\         
|- log (2*x)*sin(x) - ------------------------ + -----------------|*log(2*x)
|                                 2                      x        |         
\                                x                                /         
$$\left(- \log{\left(2 x \right)}^{2} \sin{\left(x \right)} + \frac{6 \log{\left(2 x \right)} \cos{\left(x \right)}}{x} - \frac{3 \left(\log{\left(2 x \right)} - 2\right) \sin{\left(x \right)}}{x^{2}}\right) \log{\left(2 x \right)}$$
The third derivative [src]
                          2                 /       2                  \                                           
     3               9*log (2*x)*sin(x)   6*\1 + log (2*x) - 3*log(2*x)/*sin(x)   9*(-2 + log(2*x))*cos(x)*log(2*x)
- log (2*x)*cos(x) - ------------------ + ------------------------------------- - ---------------------------------
                             x                               3                                     2               
                                                            x                                     x                
$$- \log{\left(2 x \right)}^{3} \cos{\left(x \right)} - \frac{9 \log{\left(2 x \right)}^{2} \sin{\left(x \right)}}{x} - \frac{9 \left(\log{\left(2 x \right)} - 2\right) \log{\left(2 x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{6 \left(\log{\left(2 x \right)}^{2} - 3 \log{\left(2 x \right)} + 1\right) \sin{\left(x \right)}}{x^{3}}$$
The graph
Derivative of y=sin(x)*log2x^3