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The derivative of the function/ 2+sqrt(3x+2)

Derivative of 2+sqrt(3x+2)

The solution

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      _________
2 + \/ 3*x + 2

$$\sqrt{3 x + 2} + 2$$

2 + sqrt(3*x + 2)

Detail solution

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

$\frac{3}{2 \sqrt{3 x + 2}}$

The answer is:

$\frac{3}{2 \sqrt{3 x + 2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$\frac{3}{2 \sqrt{3 x + 2}}$$

Simplify

The second derivative [src]

     -9       
--------------
           3/2
4*(2 + 3*x)

$$- \frac{9}{4 \left(3 x + 2\right)^{\frac{3}{2}}}$$

Simplify

The third derivative [src]

      81      
--------------
           5/2
8*(2 + 3*x)

$$\frac{81}{8 \left(3 x + 2\right)^{\frac{5}{2}}}$$

Simplify