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(e^(2x)(2e^x-3))/6

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Identical expressions
(e^(2x)(2e^x- three))/ six
(e to the power of (2x)(2e to the power of x minus 3)) divide by 6
(e to the power of (2x)(2e to the power of x minus three)) divide by six
(e(2x)(2ex-3))/6
e2x2ex-3/6
e^2x2e^x-3/6
(e^(2x)(2e^x-3)) divide by 6
Similar expressions
(e^(2x)(2e^x+3))/6

The derivative of the function/ (e^(2x)(2e^x-3))/6

Derivative of (e^(2x)(2e^x-3))/6

The solution

You have entered [src]

 2*x /   x    \
e   *\2*e  - 3/
---------------
       6

$$\frac{\left(2 e^{x} - 3\right) e^{2 x}}{6}$$

  / 2*x /   x    \\
d |e   *\2*e  - 3/|
--|---------------|
dx\       6       /

$$\frac{d}{d x} \frac{\left(2 e^{x} - 3\right) e^{2 x}}{6}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. 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:
    
    $2 e^{2 x}$
  $g{\left(x \right)} = 2 e^{x} - 3$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $2 e^{x} - 3$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. The derivative of $e^{x}$ is itself.
      So, the result is: $2 e^{x}$
    2. The derivative of the constant $\left(-1\right) 3$ is zero.
    The result is: $2 e^{x}$
  The result is: $2 \cdot \left(2 e^{x} - 3\right) e^{2 x} + 2 e^{3 x}$
So, the result is: $\frac{\left(2 e^{x} - 3\right) e^{2 x}}{3} + \frac{e^{3 x}}{3}$
Now simplify:

$\left(e^{x} - 1\right) e^{2 x}$

The answer is:

$\left(e^{x} - 1\right) e^{2 x}$

The graph

Plot the graph f(x)

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The first derivative [src]

 3*x   /   x    \  2*x
e      \2*e  - 3/*e   
---- + ---------------
 3            3

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

Simplify

The second derivative [src]

/        x\  2*x
\-6 + 9*e /*e   
----------------
       3

$$\frac{\left(9 e^{x} - 6\right) e^{2 x}}{3}$$

Simplify

The third derivative [src]

/          x\  2*x
\-12 + 27*e /*e   
------------------
        3

$$\frac{\left(27 e^{x} - 12\right) e^{2 x}}{3}$$

Simplify

The graph

Derivative of (e^(2x)(2e^x-3))/6