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Graphing y =:
log(x)^2*sin(x)
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Similar expressions
log(x)^2*sinx

The derivative of the function/ log(x)^2*sin(x)

Derivative of log(x)^2*sin(x)

The solution

You have entered [src]

   2          
log (x)*sin(x)

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

d /   2          \
--\log (x)*sin(x)/
dx

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

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \log{\left(x \right)}^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \log{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
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:
  
  $\frac{2 \log{\left(x \right)}}{x}$
$g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $\log{\left(x \right)}^{2} \cos{\left(x \right)} + \frac{2 \log{\left(x \right)} \sin{\left(x \right)}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

     2             2*(-1 + log(x))*sin(x)   4*cos(x)*log(x)
- log (x)*sin(x) - ---------------------- + ---------------
                              2                    x       
                             x

$$- \log{\left(x \right)}^{2} \sin{\left(x \right)} + \frac{4 \log{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{2 \left(\log{\left(x \right)} - 1\right) \sin{\left(x \right)}}{x^{2}}$$

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of log(x)^2*sin(x)

The graph:

Piecewise:

The solution