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Derivative of:
Derivative of e^(2-x)
Derivative of x^(4/5)
Derivative of f(x)=5
Derivative of 2/x²
Integral of d{x}:
e^(x*(-2))*cos(3*x)
Identical expressions
e^(x*(- two))*cos(three *x)
e to the power of (x multiply by ( minus 2)) multiply by co sinus of e of (3 multiply by x)
e to the power of (x multiply by ( minus two)) multiply by co sinus of e of (three multiply by x)
e(x*(-2))*cos(3*x)
ex*-2*cos3*x
e^(x(-2))cos(3x)
e(x(-2))cos(3x)
ex-2cos3x
e^x-2cos3x
Similar expressions
e^(x*(2))*cos(3*x)

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

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

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

E^(x*(-2))*cos(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:
  
  $- 2 e^{\left(-2\right) x}$
$g{\left(x \right)} = \cos{\left(3 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
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 is: $- 3 e^{\left(-2\right) x} \sin{\left(3 x \right)} - 2 e^{\left(-2\right) x} \cos{\left(3 x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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