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Identical expressions
(xexp^(2x)+ three)^ five
(x exponent of to the power of (2x) plus 3) to the power of 5
(x exponent of to the power of (2x) plus three) to the power of five
(xexp(2x)+3)5
xexp2x+35
(xexp^(2x)+3)⁵
xexp^2x+3^5
Similar expressions
(xexp^(2x)-3)^5

The derivative of the function/ (xexp^(2x)+3)^5

Derivative of (xexp^(2x)+3)^5

The solution

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            5
/   2*x    \ 
\x*E    + 3/

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

(x*E^(2*x) + 3)^5

Detail solution

Let $u = e^{2 x} x + 3$ .
Apply the power rule: $u^{5}$ goes to $5 u^{4}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(e^{2 x} x + 3\right)$ :
1. Differentiate $e^{2 x} x + 3$ term by term:
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the power rule: $x$ goes to $1$
    $g{\left(x \right)} = e^{2 x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    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.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
      The result of the chain rule is:
      
      $2 e^{2 x}$
    The result is: $2 x e^{2 x} + e^{2 x}$
  2. The derivative of the constant $3$ is zero.
  The result is: $2 x e^{2 x} + e^{2 x}$
The result of the chain rule is:

$5 \left(e^{2 x} x + 3\right)^{4} \left(2 x e^{2 x} + e^{2 x}\right)$
Now simplify:

$\left(10 x + 5\right) \left(x e^{2 x} + 3\right)^{4} e^{2 x}$

The answer is:

$\left(10 x + 5\right) \left(x e^{2 x} + 3\right)^{4} e^{2 x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

            4                     
/   2*x    \  /   2*x         2*x\
\x*E    + 3/ *\5*e    + 10*x*e   /

$$\left(e^{2 x} x + 3\right)^{4} \left(10 x e^{2 x} + 5 e^{2 x}\right)$$

Simplify

The second derivative [src]

               3                                              
   /       2*x\  /         2  2*x           /       2*x\\  2*x
20*\3 + x*e   / *\(1 + 2*x) *e    + (1 + x)*\3 + x*e   //*e

$$20 \left(x e^{2 x} + 3\right)^{3} \left(\left(x + 1\right) \left(x e^{2 x} + 3\right) + \left(2 x + 1\right)^{2} e^{2 x}\right) e^{2 x}$$

Simplify

The third derivative [src]

               2 /            2                                                                       \     
   /       2*x\  |/       2*x\                         3  4*x                        /       2*x\  2*x|  2*x
20*\3 + x*e   / *\\3 + x*e   / *(3 + 2*x) + 3*(1 + 2*x) *e    + 12*(1 + x)*(1 + 2*x)*\3 + x*e   /*e   /*e

$$20 \left(x e^{2 x} + 3\right)^{2} \left(12 \left(x + 1\right) \left(2 x + 1\right) \left(x e^{2 x} + 3\right) e^{2 x} + 3 \left(2 x + 1\right)^{3} e^{4 x} + \left(2 x + 3\right) \left(x e^{2 x} + 3\right)^{2}\right) e^{2 x}$$

Simplify

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Derivative of (xexp^(2x)+3)^5

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The solution