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3*sqrt(e^(4x+3))

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Identical expressions
three *sqrt(e^(4x+ three))
3 multiply by square root of (e to the power of (4x plus 3))
three multiply by square root of (e to the power of (4x plus three))
3*√(e^(4x+3))
3*sqrt(e(4x+3))
3*sqrte4x+3
3sqrt(e^(4x+3))
3sqrt(e(4x+3))
3sqrte4x+3
3sqrte^4x+3
Similar expressions
3*sqrt(e^(4x-3))

The derivative of the function/ 3*sqrt(e^(4x+3))

Derivative of 3*sqrt(e^(4x+3))

The solution

You have entered [src]

     __________
    /  4*x + 3 
3*\/  e

$$3 \sqrt{e^{4 x + 3}}$$

  /     __________\
d |    /  4*x + 3 |
--\3*\/  e        /
dx

$$\frac{d}{d x} 3 \sqrt{e^{4 x + 3}}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = e^{4 x + 3}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} e^{4 x + 3}$ :
  1. Let $u = 4 x + 3$ .
  2. The derivative of $e^{u}$ is itself.
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 x + 3\right)$ :
    1. Differentiate $4 x + 3$ 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: $4$
      2. The derivative of the constant $3$ is zero.
      The result is: $4$
    The result of the chain rule is:
    
    $4 e^{4 x + 3}$
  The result of the chain rule is:
  
  $2 e^{- 2 x - \frac{3}{2}} e^{4 x + 3}$
So, the result is: $6 e^{- 2 x - \frac{3}{2}} e^{4 x + 3}$
Now simplify:

$6 e^{2 x + \frac{3}{2}}$

The answer is:

$6 e^{2 x + \frac{3}{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   -3 - 4*x  3/2 + 2*x  4*x + 3
6*e        *e         *e

$$6 e^{- 4 x - 3} e^{2 x + \frac{3}{2}} e^{4 x + 3}$$

Simplify

The second derivative [src]

    3/2 + 2*x
12*e

$$12 e^{2 x + \frac{3}{2}}$$

Simplify

The third derivative [src]

    3/2 + 2*x
24*e

$$24 e^{2 x + \frac{3}{2}}$$

Simplify

The graph

Derivative of 3*sqrt(e^(4x+3))