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y=sin(x)*log2x^3
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  • Identical expressions

  • y=sin(x)*log2x^ three
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  • Similar expressions

  • y=sinx*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(2x)3sin(x)\log{\left(2 x \right)}^{3} \sin{\left(x \right)}
sin(x)*log(2*x)^3
Detail solution
  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=sin(x)f{\left(x \right)} = \sin{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of sine is cosine:

      ddxsin(x)=cos(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

    g(x)=log(2x)3g{\left(x \right)} = \log{\left(2 x \right)}^{3}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=log(2x)u = \log{\left(2 x \right)}.

    2. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

    3. Then, apply the chain rule. Multiply by ddxlog(2x)\frac{d}{d x} \log{\left(2 x \right)}:

      1. Let u=2xu = 2 x.

      2. The derivative of log(u)\log{\left(u \right)} is 1u\frac{1}{u}.

      3. Then, apply the chain rule. Multiply by ddx2x\frac{d}{d x} 2 x:

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

          1. Apply the power rule: xx goes to 11

          So, the result is: 22

        The result of the chain rule is:

        1x\frac{1}{x}

      The result of the chain rule is:

      3log(2x)2x\frac{3 \log{\left(2 x \right)}^{2}}{x}

    The result is: log(2x)3cos(x)+3log(2x)2sin(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}

  2. Now simplify:

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


The answer is:

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

The graph
02468-8-6-4-2-1010-5050
The first derivative [src]
                        2            
   3               3*log (2*x)*sin(x)
log (2*x)*cos(x) + ------------------
                           x         
log(2x)3cos(x)+3log(2x)2sin(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                                /         
(log(2x)2sin(x)+6log(2x)cos(x)x3(log(2x)2)sin(x)x2)log(2x)\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(2x)3cos(x)9log(2x)2sin(x)x9(log(2x)2)log(2x)cos(x)x2+6(log(2x)23log(2x)+1)sin(x)x3- \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