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Derivative of:
Derivative of x*sin(2*x)
Derivative of x^4/(x^3-1)
Derivative of (x^4-x-1)^4
Derivative of x^4*cos(x)
Identical expressions
y=sin^ two (cos3x)
y equally sinus of squared ( co sinus of e of 3x)
y equally sinus of to the power of two ( co sinus of e of 3x)
y=sin2(cos3x)
y=sin2cos3x
y=sin²(cos3x)
y=sin to the power of 2(cos3x)
y=sin^2cos3x

The derivative of the function/ y=sin^2(cos3x)

Derivative of y=sin^2(cos3x)

The solution

You have entered [src]

   2          
sin (cos(3*x))

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

sin(cos(3*x))^2

Detail solution

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

$- 6 \sin{\left(3 x \right)} \sin{\left(\cos{\left(3 x \right)} \right)} \cos{\left(\cos{\left(3 x \right)} \right)}$
Now simplify:

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

The answer is:

- \frac{3 \cos{\left(3 x - 2 \cos{\left(3 x \right)} \right)}}{2} + \frac{3 \cos{\left(3 x + 2 \cos{\left(3 x \right)} \right)}}{2}

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-6*cos(cos(3*x))*sin(3*x)*sin(cos(3*x))

- 6 \sin{\left(3 x \right)} \sin{\left(\cos{\left(3 x \right)} \right)} \cos{\left(\cos{\left(3 x \right)} \right)}

Simplify

The second derivative [src]

   /   2              2           2         2                                                 \
18*\cos (cos(3*x))*sin (3*x) - sin (3*x)*sin (cos(3*x)) - cos(3*x)*cos(cos(3*x))*sin(cos(3*x))/

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

Simplify

The third derivative [src]

   /                                   2                           2                           2                                 \         
54*\cos(cos(3*x))*sin(cos(3*x)) - 3*sin (cos(3*x))*cos(3*x) + 3*cos (cos(3*x))*cos(3*x) + 4*sin (3*x)*cos(cos(3*x))*sin(cos(3*x))/*sin(3*x)

54 \left(4 \sin^{2}{\left(3 x \right)} \sin{\left(\cos{\left(3 x \right)} \right)} \cos{\left(\cos{\left(3 x \right)} \right)} - 3 \sin^{2}{\left(\cos{\left(3 x \right)} \right)} \cos{\left(3 x \right)} + \sin{\left(\cos{\left(3 x \right)} \right)} \cos{\left(\cos{\left(3 x \right)} \right)} + 3 \cos{\left(3 x \right)} \cos^{2}{\left(\cos{\left(3 x \right)} \right)}\right) \sin{\left(3 x \right)}

Simplify

The graph

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Identical expressions

Derivative of y=sin^2(cos3x)

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The solution