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Identical expressions
y=x^ two *ln(sinx)
y equally x squared multiply by ln( sinus of x)
y equally x to the power of two multiply by ln( sinus of x)
y=x2*ln(sinx)
y=x2*lnsinx
y=x²*ln(sinx)
y=x to the power of 2*ln(sinx)
y=x^2ln(sinx)
y=x2ln(sinx)
y=x2lnsinx
y=x^2lnsinx
Similar expressions
e^(2x^2)*ln(sin(x))

The derivative of the function/ y=x^2*ln(sinx)

Derivative of y=x^2*ln(sinx)

The solution

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 2            
x *log(sin(x))

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

x^2*log(sin(x))

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                                    /       2   \       \
  |                                  2 |    cos (x)|       |
  |                                 x *|1 + -------|*cos(x)|
  |      /       2   \                 |       2   |       |
  |      |    cos (x)|   3*cos(x)      \    sin (x)/       |
2*|- 3*x*|1 + -------| + -------- + -----------------------|
  |      |       2   |    sin(x)             sin(x)        |
  \      \    sin (x)/                                     /

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

Simplify

The graph

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Derivative of y=x^2*ln(sinx)

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The solution