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Derivative of:
Derivative of e^x^x
Derivative of sqrt4
Derivative of cos(x)/e^x
Derivative of acos(1/x)
Identical expressions
exp((sin(three *x))^ two)
exponent of (( sinus of (3 multiply by x)) squared )
exponent of (( sinus of (three multiply by x)) to the power of two)
exp((sin(3*x))2)
expsin3*x2
exp((sin(3*x))²)
exp((sin(3*x)) to the power of 2)
exp((sin(3x))^2)
exp((sin(3x))2)
expsin3x2
expsin3x^2

The derivative of the function/ exp((sin(3*x))^2)

Derivative of exp((sin(3*x))^2)

The solution

You have entered [src]

    2     
 sin (3*x)
e

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

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

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

Transform

Detail solution

Let $u = \sin^{2}{\left(3 x \right)}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin^{2}{\left(3 x \right)}$ :
1. Let $u = \sin{\left(3 x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
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.
      1. 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:
  
  $6 \sin{\left(3 x \right)} \cos{\left(3 x \right)}$
The result of the chain rule is:

$6 e^{\sin^{2}{\left(3 x \right)}} \sin{\left(3 x \right)} \cos{\left(3 x \right)}$
Now simplify:

$3 e^{\frac{1}{2} - \frac{\cos{\left(6 x \right)}}{2}} \sin{\left(6 x \right)}$

The answer is:

$3 e^{\frac{1}{2} - \frac{\cos{\left(6 x \right)}}{2}} \sin{\left(6 x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

               2              
            sin (3*x)         
6*cos(3*x)*e         *sin(3*x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                                          2              
    /          2             2             2         2     \           sin (3*x)         
108*\-2 - 3*sin (3*x) + 3*cos (3*x) + 2*cos (3*x)*sin (3*x)/*cos(3*x)*e         *sin(3*x)

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

Simplify

The graph

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Identical expressions

Derivative of exp((sin(3*x))^2)

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The solution