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Derivative of e^(sin(1-3*x)^(2))
Graphing y =:
e^(sin(1-3*x)^(2))
Identical expressions
e^(sin(one - three *x)^(two))
e to the power of ( sinus of (1 minus 3 multiply by x) to the power of (2))
e to the power of ( sinus of (one minus three multiply by x) to the power of (two))
e(sin(1-3*x)(2))
esin1-3*x2
e^(sin(1-3x)^(2))
e(sin(1-3x)(2))
esin1-3x2
e^sin1-3x^2
Similar expressions
e^(sin(1+3*x)^(2))

The derivative of the function/ e^(sin(1-3*x)^(2))

Derivative of e^(sin(1-3*x)^(2))

The solution

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    2         
 sin (1 - 3*x)
E

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

E^(sin(1 - 3*x)^2)

Detail solution

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

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

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

The answer is:

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

The graph

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The first derivative [src]

                     2                      
                  sin (1 - 3*x)             
-6*cos(-1 + 3*x)*e             *sin(1 - 3*x)

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

Simplify

The second derivative [src]

                                                                           2          
   /   2                2                  2              2          \  sin (-1 + 3*x)
18*\cos (-1 + 3*x) - sin (-1 + 3*x) + 2*cos (-1 + 3*x)*sin (-1 + 3*x)/*e

$$18 \left(2 \sin^{2}{\left(3 x - 1 \right)} \cos^{2}{\left(3 x - 1 \right)} - \sin^{2}{\left(3 x - 1 \right)} + \cos^{2}{\left(3 x - 1 \right)}\right) e^{\sin^{2}{\left(3 x - 1 \right)}}$$

Simplify

The third derivative [src]

                                                                                                   2                        
    /          2                  2                  2              2          \                sin (-1 + 3*x)              
108*\-2 - 3*sin (-1 + 3*x) + 3*cos (-1 + 3*x) + 2*cos (-1 + 3*x)*sin (-1 + 3*x)/*cos(-1 + 3*x)*e              *sin(-1 + 3*x)

$$108 \left(2 \sin^{2}{\left(3 x - 1 \right)} \cos^{2}{\left(3 x - 1 \right)} - 3 \sin^{2}{\left(3 x - 1 \right)} + 3 \cos^{2}{\left(3 x - 1 \right)} - 2\right) e^{\sin^{2}{\left(3 x - 1 \right)}} \sin{\left(3 x - 1 \right)} \cos{\left(3 x - 1 \right)}$$

Simplify

The graph

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Derivative of e^(sin(1-3*x)^(2))

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The solution