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Identical expressions
y=x^2ln(e^sin(3x))
y equally x squared ln(e to the power of sinus of (3x))
y=x2ln(esin(3x))
y=x2lnesin3x
y=x²ln(e^sin(3x))
y=x to the power of 2ln(e to the power of sin(3x))
y=x^2lne^sin3x

The derivative of the function/ y=x^2ln(e^sin(3x))

Derivative of y=x^2ln(e^sin(3x))

The solution

You have entered [src]

 2    / sin(3*x)\
x *log\e        /

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

d / 2    / sin(3*x)\\
--\x *log\e        //
dx

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

Transform

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       / sin(3*x)\      2         
2*x*log\e        / + 3*x *cos(3*x)

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

Simplify

The second derivative [src]

                2                         
2*sin(3*x) - 9*x *sin(3*x) + 12*x*cos(3*x)

$$- 9 x^{2} \sin{\left(3 x \right)} + 12 x \cos{\left(3 x \right)} + 2 \sin{\left(3 x \right)}$$

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Identical expressions

Derivative of y=x^2ln(e^sin(3x))

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The solution