Derivative of y=sin(3x²)cos(3x²)

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sin\3*x /*cos\3*x /

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

sin(3*x^2)*cos(3*x^2)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin{\left(3 x^{2} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 3 x^{2}$ .
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 $\frac{d}{d x} 3 x^{2}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $6 x$
  The result of the chain rule is:
  
  $6 x \cos{\left(3 x^{2} \right)}$
$g{\left(x \right)} = \cos{\left(3 x^{2} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 3 x^{2}$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x^{2}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $6 x$
  The result of the chain rule is:
  
  $- 6 x \sin{\left(3 x^{2} \right)}$
The result is: $- 6 x \sin^{2}{\left(3 x^{2} \right)} + 6 x \cos^{2}{\left(3 x^{2} \right)}$
Now simplify:

$6 x \cos{\left(6 x^{2} \right)}$

The answer is:

$6 x \cos{\left(6 x^{2} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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- 6*x*sin \3*x / + 6*x*cos \3*x /

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

Simplify

The second derivative [src]

   //     /   2\      2    /   2\\    /   2\   /   2    /   2\      /   2\\    /   2\       2    /   2\    /   2\\
-6*\\- cos\3*x / + 6*x *sin\3*x //*cos\3*x / + \6*x *cos\3*x / + sin\3*x //*sin\3*x / + 12*x *cos\3*x /*sin\3*x //

$$- 6 \left(12 x^{2} \sin{\left(3 x^{2} \right)} \cos{\left(3 x^{2} \right)} + \left(6 x^{2} \sin{\left(3 x^{2} \right)} - \cos{\left(3 x^{2} \right)}\right) \cos{\left(3 x^{2} \right)} + \left(6 x^{2} \cos{\left(3 x^{2} \right)} + \sin{\left(3 x^{2} \right)}\right) \sin{\left(3 x^{2} \right)}\right)$$

Simplify

The third derivative [src]

      //     /   2\      2    /   2\\    /   2\   /     /   2\      2    /   2\\    /   2\   /   2    /   2\      /   2\\    /   2\   /   2    /   2\      /   2\\    /   2\\
108*x*\\- cos\3*x / + 2*x *sin\3*x //*sin\3*x / + \- cos\3*x / + 6*x *sin\3*x //*sin\3*x / - \2*x *cos\3*x / + sin\3*x //*cos\3*x / - \6*x *cos\3*x / + sin\3*x //*cos\3*x //

$$108 x \left(\left(2 x^{2} \sin{\left(3 x^{2} \right)} - \cos{\left(3 x^{2} \right)}\right) \sin{\left(3 x^{2} \right)} + \left(6 x^{2} \sin{\left(3 x^{2} \right)} - \cos{\left(3 x^{2} \right)}\right) \sin{\left(3 x^{2} \right)} - \left(2 x^{2} \cos{\left(3 x^{2} \right)} + \sin{\left(3 x^{2} \right)}\right) \cos{\left(3 x^{2} \right)} - \left(6 x^{2} \cos{\left(3 x^{2} \right)} + \sin{\left(3 x^{2} \right)}\right) \cos{\left(3 x^{2} \right)}\right)$$

Simplify

The graph

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Derivative of y=sin(3x²)cos(3x²)

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The solution