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Identical expressions
y=ln^ three *sinx
y equally ln cubed multiply by sinus of x
y equally ln to the power of three multiply by sinus of x
y=ln3*sinx
y=ln³*sinx
y=ln to the power of 3*sinx
y=ln^3sinx
y=ln3sinx

The derivative of the function/ y=ln^3*sinx

Derivative of y=ln^3*sinx

The solution

You have entered [src]

   3          
log (x)*sin(x)

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

log(x)^3*sin(x)

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

/     2             3*(-2 + log(x))*sin(x)   6*cos(x)*log(x)\       
|- log (x)*sin(x) - ---------------------- + ---------------|*log(x)
|                              2                    x       |       
\                             x                             /

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

Simplify

The third derivative [src]

                        2               /       2              \                                       
     3             9*log (x)*sin(x)   6*\1 + log (x) - 3*log(x)/*sin(x)   9*(-2 + log(x))*cos(x)*log(x)
- log (x)*cos(x) - ---------------- + --------------------------------- - -----------------------------
                          x                            3                                 2             
                                                      x                                 x

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

Simplify

The graph

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Derivative of y=ln^3*sinx

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The solution