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Similar expressions
y=sinx*log2x^3

The derivative of the function/ y=sin(x)*log2x^3

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

The solution

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          3     
sin(x)*log (2*x)

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

sin(x)*log(2*x)^3

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
$g{\left(x \right)} = \log{\left(2 x \right)}^{3}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \log{\left(2 x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(2 x \right)}$ :
  1. Let $u = 2 x$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2$
    The result of the chain rule is:
    
    $\frac{1}{x}$
  The result of the chain rule is:
  
  $\frac{3 \log{\left(2 x \right)}^{2}}{x}$
The result is: $\log{\left(2 x \right)}^{3} \cos{\left(x \right)} + \frac{3 \log{\left(2 x \right)}^{2} \sin{\left(x \right)}}{x}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

                        2            
   3               3*log (2*x)*sin(x)
log (2*x)*cos(x) + ------------------
                           x

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

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                                /

$$\left(- \log{\left(2 x \right)}^{2} \sin{\left(x \right)} + \frac{6 \log{\left(2 x \right)} \cos{\left(x \right)}}{x} - \frac{3 \left(\log{\left(2 x \right)} - 2\right) \sin{\left(x \right)}}{x^{2}}\right) \log{\left(2 x \right)}$$

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

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

Simplify

The graph

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

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The solution