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Derivative of:
Derivative of x^12
Derivative of e^(2-x)
Derivative of 5e^x
Derivative of 3sinx
Identical expressions
e^(x*(- two))*sin(three *x)
e to the power of (x multiply by ( minus 2)) multiply by sinus of (3 multiply by x)
e to the power of (x multiply by ( minus two)) multiply by sinus of (three multiply by x)
e(x*(-2))*sin(3*x)
ex*-2*sin3*x
e^(x(-2))sin(3x)
e(x(-2))sin(3x)
ex-2sin3x
e^x-2sin3x
Similar expressions
e^(x*(2))*sin(3*x)

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

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

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

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

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = e^{\left(-2\right) x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \left(-2\right) x$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(-2\right) x$ :
  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: $-2$
  The result of the chain rule is:
  
  $- 2 e^{\left(-2\right) x}$
$g{\left(x \right)} = \sin{\left(3 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
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 is: $- 2 e^{\left(-2\right) x} \sin{\left(3 x \right)} + 3 e^{\left(-2\right) x} \cos{\left(3 x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                             -2*x
(-12*cos(3*x) - 5*sin(3*x))*e

\left(- 5 \sin{\left(3 x \right)} - 12 \cos{\left(3 x \right)}\right) e^{- 2 x}

Simplify

The third derivative [src]

                            -2*x
(9*cos(3*x) + 46*sin(3*x))*e

\left(46 \sin{\left(3 x \right)} + 9 \cos{\left(3 x \right)}\right) e^{- 2 x}

Simplify

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