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e^(two *x+ two)*(four *x+ two)/ four *(x+ one)^ two
e to the power of (2 multiply by x plus 2) multiply by (4 multiply by x plus 2) divide by 4 multiply by (x plus 1) squared
e to the power of (two multiply by x plus two) multiply by (four multiply by x plus two) divide by four multiply by (x plus one) to the power of two
e(2*x+2)*(4*x+2)/4*(x+1)2
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e to the power of (2*x+2)*(4*x+2)/4*(x+1) to the power of 2
e^(2x+2)(4x+2)/4(x+1)^2
e(2x+2)(4x+2)/4(x+1)2
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e^(2*x+2)*(4*x+2) divide by 4*(x+1)^2
Similar expressions
e^(2*x-2)*(4*x+2)/4*(x+1)^2
e^(2*x+2)*(4*x-2)/4*(x+1)^2
e^(2*x+2)*(4*x+2)/4*(x-1)^2

The derivative of the function/ e^(2*x+2)*(4*x+2)/4*(x+1)^2

Derivative of e^(2x+2)(4x+2)/4(x+1)^2

The solution

You have entered [src]

 2*x + 2                   
E       *(4*x + 2)        2
------------------*(x + 1) 
        4

$$\frac{e^{2 x + 2} \left(4 x + 2\right)}{4} \left(x + 1\right)^{2}$$

((E^(2*x + 2)*(4*x + 2))/4)*(x + 1)^2

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \left(x + 1\right)^{2} \left(4 x + 2\right) e^{2 x + 2}$ and $g{\left(x \right)} = 4$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} h{\left(x \right)} = f{\left(x \right)} g{\left(x \right)} \frac{d}{d x} h{\left(x \right)} + f{\left(x \right)} h{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} h{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \left(x + 1\right)^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = x + 1$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)$ :
    1. Differentiate $x + 1$ term by term:
      1. The derivative of the constant $1$ is zero.
      2. Apply the power rule: $x$ goes to $1$
      The result is: $1$
    The result of the chain rule is:
    
    $2 x + 2$
  $g{\left(x \right)} = 4 x + 2$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $4 x + 2$ term by term:
    1. The derivative of the constant $2$ is zero.
    2. 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 is: $4$
  $h{\left(x \right)} = e^{2 x + 2}$ ; to find $\frac{d}{d x} h{\left(x \right)}$ :
  1. Let $u = 2 x + 2$ .
  2. The derivative of $e^{u}$ is itself.
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 2\right)$ :
    1. Differentiate $2 x + 2$ term by term:
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
      2. The derivative of the constant $2$ is zero.
      The result is: $2$
    The result of the chain rule is:
    
    $2 e^{2 x + 2}$
  The result is: $2 \left(x + 1\right)^{2} \left(4 x + 2\right) e^{2 x + 2} + 4 \left(x + 1\right)^{2} e^{2 x + 2} + \left(2 x + 2\right) \left(4 x + 2\right) e^{2 x + 2}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of the constant $4$ is zero.
Now plug in to the quotient rule:

$\frac{\left(x + 1\right)^{2} \left(4 x + 2\right) e^{2 x + 2}}{2} + \left(x + 1\right)^{2} e^{2 x + 2} + \frac{\left(2 x + 2\right) \left(4 x + 2\right) e^{2 x + 2}}{4}$
Now simplify:

$\left(x + 1\right) \left(3 x + \left(x + 1\right) \left(2 x + 1\right) + 2\right) e^{2 x + 2}$

The answer is:

$\left(x + 1\right) \left(3 x + \left(x + 1\right) \left(2 x + 1\right) + 2\right) e^{2 x + 2}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         /           2*x + 2           \                        2*x + 2
       2 |(4*x + 2)*e           2*x + 2|   (2 + 2*x)*(4*x + 2)*e       
(x + 1) *|------------------ + e       | + ----------------------------
         \        2                    /                4

$$\left(x + 1\right)^{2} \left(\frac{\left(4 x + 2\right) e^{2 x + 2}}{2} + e^{2 x + 2}\right) + \frac{\left(2 x + 2\right) \left(4 x + 2\right) e^{2 x + 2}}{4}$$

Simplify

The second derivative [src]

           2 + 2*x             /           2 + 2*x    2*(1 + x)\            2            2 + 2*x
(1 + 2*x)*e        + 4*(1 + x)*\(1 + 2*x)*e        + e         / + 2*(1 + x) *(3 + 2*x)*e

$$2 \left(x + 1\right)^{2} \left(2 x + 3\right) e^{2 x + 2} + 4 \left(x + 1\right) \left(\left(2 x + 1\right) e^{2 x + 2} + e^{2 \left(x + 1\right)}\right) + \left(2 x + 1\right) e^{2 x + 2}$$

Simplify

The third derivative [src]

  /                   2                              \  2 + 2*x
2*\6 + 6*x + 4*(1 + x) *(2 + x) + 6*(1 + x)*(3 + 2*x)/*e

$$2 \left(6 x + 4 \left(x + 1\right)^{2} \left(x + 2\right) + 6 \left(x + 1\right) \left(2 x + 3\right) + 6\right) e^{2 x + 2}$$

Simplify

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Derivative of e^(2x+2)(4x+2)/4(x+1)^2

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of e^(2*x+2)*(4*x+2)/4*(x+1)^2

The graph:

Piecewise:

The solution

Derivative of e^(2x+2)(4x+2)/4(x+1)^2