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Derivative of:
Derivative of (5*x-4)*(2*x^4-7*x+1)
Derivative of x/5+5/x
Derivative of (x+5)^2*(x-1)+7
Derivative of x^3-48x+17
Integral of d{x}:
e^(2*x)*sin(3*x)
Identical expressions
e^(two *x)*sin(three *x)
e to the power of (2 multiply by x) multiply by sinus of (3 multiply by x)
e to the power of (two multiply by x) multiply by sinus of (three multiply by x)
e(2*x)*sin(3*x)
e2*x*sin3*x
e^(2x)sin(3x)
e(2x)sin(3x)
e2xsin3x
e^2xsin3x

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

You have entered [src]

 2*x         
E   *sin(3*x)

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

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

Detail solution

Apply the product rule:

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

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

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

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

Simplify

The second derivative [src]

                             2*x
(-5*sin(3*x) + 12*cos(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
(-46*sin(3*x) + 9*cos(3*x))*e

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

Simplify

The graph

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