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y=sin(x)*log2x^3

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  • Derivative of:
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  • Derivative of x^3/x Derivative of x^3/x
  • Derivative of x^2+4 Derivative of x^2+4
  • Derivative of (x^2-1)^3 Derivative of (x^2-1)^3

  • Identical expressions

  • y=sin(x)*log2x^ three
  • y equally sinus of (x) multiply by logarithm of 2x cubed
  • y equally sinus of (x) multiply by logarithm of 2x to the power of three
  • y=sin(x)*log2x3
  • y=sinx*log2x3
  • y=sin(x)*log2x³
  • y=sin(x)*log2x to the power of 3
  • y=sin(x)log2x^3
  • y=sin(x)log2x3
  • y=sinxlog2x3
  • y=sinxlog2x^3

  • Similar expressions

  • y=sinx*log2x^3
The derivative of the function/ 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⁡(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
Plot the graph f(x)
Plot the graph f'(x)
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}
Simplify
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)x−3(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)}
Simplify
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)x−9(log⁡(2x)−2)log⁡(2x)cos⁡(x)x2+6(log⁡(2x)2−3log⁡(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}}
Simplify
The graph
Derivative of y=sin(x)*log2x^3
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