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Derivative of y=root5(3x+1)^2

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$$\left(3 x + 1\right)^{\left(\frac{1}{5}\right)^{2}}$$

  /         1 \
  |         --|
  |          2|
d |         5 |
--\(3*x + 1)  /
dx

$$\frac{d}{d x} \left(3 x + 1\right)^{\left(\frac{1}{5}\right)^{2}}$$

Transform

Detail solution

Let $u = 3 x + 1$ .
Apply the power rule: $u^{\left(\frac{1}{5}\right)^{2}}$ goes to $\frac{u^{\left(\frac{1}{5}\right)^{2}}}{25 u}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x + 1\right)$ :
1. Differentiate $3 x + 1$ 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 $1$ is zero.
  The result is: $3$
The result of the chain rule is:

$\frac{3 \left(3 x + 1\right)^{\left(\frac{1}{5}\right)^{2}}}{25 \cdot \left(3 x + 1\right)}$
Now simplify:

$\frac{3}{25 \left(3 x + 1\right)^{\frac{24}{25}}}$

The answer is:

$\frac{3}{25 \left(3 x + 1\right)^{\frac{24}{25}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

           1 
           --
            2
           5 
3*(3*x + 1)  
-------------
 25*(3*x + 1)

$$\frac{3 \left(3 x + 1\right)^{\left(\frac{1}{5}\right)^{2}}}{25 \cdot \left(3 x + 1\right)}$$

Simplify

The second derivative [src]

     -216      
---------------
             49
             --
             25
625*(1 + 3*x)

$$- \frac{216}{625 \left(3 x + 1\right)^{\frac{49}{25}}}$$

Simplify

The third derivative [src]

      31752      
-----------------
               74
               --
               25
15625*(1 + 3*x)

$$\frac{31752}{15625 \left(3 x + 1\right)^{\frac{74}{25}}}$$

Simplify

The graph