Derivative of y=x^3sinx*lnx

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

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

d / 3              \
--\x *sin(x)*log(x)/
dx

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

Transform

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /                                           2                                  \
x*\5*sin(x) + 2*x*cos(x) + 6*log(x)*sin(x) - x *log(x)*sin(x) + 6*x*cos(x)*log(x)/

$$x \left(- x^{2} \log{\left(x \right)} \sin{\left(x \right)} + 6 x \log{\left(x \right)} \cos{\left(x \right)} + 2 x \cos{\left(x \right)} + 6 \log{\left(x \right)} \sin{\left(x \right)} + 5 \sin{\left(x \right)}\right)$$

Simplify

3-th derivative [src]

               2                                           3                    2                                   
11*sin(x) - 3*x *sin(x) + 6*log(x)*sin(x) + 15*x*cos(x) - x *cos(x)*log(x) - 9*x *log(x)*sin(x) + 18*x*cos(x)*log(x)

$$- x^{3} \log{\left(x \right)} \cos{\left(x \right)} - 9 x^{2} \log{\left(x \right)} \sin{\left(x \right)} - 3 x^{2} \sin{\left(x \right)} + 18 x \log{\left(x \right)} \cos{\left(x \right)} + 15 x \cos{\left(x \right)} + 6 \log{\left(x \right)} \sin{\left(x \right)} + 11 \sin{\left(x \right)}$$

Simplify

The third derivative [src]

               2                                           3                    2                                   
11*sin(x) - 3*x *sin(x) + 6*log(x)*sin(x) + 15*x*cos(x) - x *cos(x)*log(x) - 9*x *log(x)*sin(x) + 18*x*cos(x)*log(x)

$$- x^{3} \log{\left(x \right)} \cos{\left(x \right)} - 9 x^{2} \log{\left(x \right)} \sin{\left(x \right)} - 3 x^{2} \sin{\left(x \right)} + 18 x \log{\left(x \right)} \cos{\left(x \right)} + 15 x \cos{\left(x \right)} + 6 \log{\left(x \right)} \sin{\left(x \right)} + 11 \sin{\left(x \right)}$$

Simplify

The graph

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Derivative of y=x^3sinx*lnx

The graph:

Piecewise:

The solution