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Derivative of:
Derivative of -6/x
Derivative of -2*y
Derivative of ln(x)+x
Derivative of cosx-3x
Identical expressions
e^(four *x)*sin(four *x)
e to the power of (4 multiply by x) multiply by sinus of (4 multiply by x)
e to the power of (four multiply by x) multiply by sinus of (four multiply by x)
e(4*x)*sin(4*x)
e4*x*sin4*x
e^(4x)sin(4x)
e(4x)sin(4x)
e4xsin4x
e^4xsin4x

Derivative of e^(4x)sin(4*x)

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 4*x         
E   *sin(4*x)

$$e^{4 x} \sin{\left(4 x \right)}$$

E^(4*x)*sin(4*x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = e^{4 x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 4 x$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$ :
  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: $4$
  The result of the chain rule is:
  
  $4 e^{4 x}$
$g{\left(x \right)} = \sin{\left(4 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 4 x$ .
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 $\frac{d}{d x} 4 x$ :
  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: $4$
  The result of the chain rule is:
  
  $4 \cos{\left(4 x \right)}$
The result is: $4 e^{4 x} \sin{\left(4 x \right)} + 4 e^{4 x} \cos{\left(4 x \right)}$
Now simplify:

$4 \left(\sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{4 x}$

The answer is:

$4 \left(\sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{4 x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

            4*x      4*x         
4*cos(4*x)*e    + 4*e   *sin(4*x)

$$4 e^{4 x} \sin{\left(4 x \right)} + 4 e^{4 x} \cos{\left(4 x \right)}$$

Simplify

The second derivative [src]

             4*x
32*cos(4*x)*e

$$32 e^{4 x} \cos{\left(4 x \right)}$$

Simplify

The third derivative [src]

                            4*x
128*(-sin(4*x) + cos(4*x))*e

$$128 \left(- \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{4 x}$$

Simplify

Derivative of e^(4*x)*sin(4*x)