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Derivative of:
Derivative of x*sqrt(3-x)
Derivative of x^6/6
Derivative of x^2+4x
Derivative of (x+1)/(x^2+1)
Integral of d{x}:
e^(3*x)*sin(2*x)
Identical expressions
e^(three *x)*sin(two *x)
e to the power of (3 multiply by x) multiply by sinus of (2 multiply by x)
e to the power of (three multiply by x) multiply by sinus of (two multiply by x)
e(3*x)*sin(2*x)
e3*x*sin2*x
e^(3x)sin(2x)
e(3x)sin(2x)
e3xsin2x
e^3xsin2x
Similar expressions
y=e^3xsin2x

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

You have entered [src]

 3*x         
E   *sin(2*x)

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

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

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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