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y=ln^5(sqrt(3x+5))

Derivative of y=ln^5(sqrt(3x+5))

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   5/  _________\
log \\/ 3*x + 5 /
$$\log{\left(\sqrt{3 x + 5} \right)}^{5}$$
log(sqrt(3*x + 5))^5
Detail solution
  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. Let .

    2. The derivative of is .

    3. Then, apply the chain rule. Multiply by :

      1. Let .

      2. Apply the power rule: goes to

      3. Then, apply the chain rule. Multiply by :

        1. Differentiate 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: goes to

            So, the result is:

          2. The derivative of the constant is zero.

          The result is:

        The result of the chain rule is:

      The result of the chain rule is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
      4/  _________\
15*log \\/ 3*x + 5 /
--------------------
    2*(3*x + 5)     
$$\frac{15 \log{\left(\sqrt{3 x + 5} \right)}^{4}}{2 \left(3 x + 5\right)}$$
The second derivative [src]
                     /       /  _________\\
      3/  _________\ |    log\\/ 5 + 3*x /|
45*log \\/ 5 + 3*x /*|1 - ----------------|
                     \           2        /
-------------------------------------------
                          2                
                 (5 + 3*x)                 
$$\frac{45 \left(1 - \frac{\log{\left(\sqrt{3 x + 5} \right)}}{2}\right) \log{\left(\sqrt{3 x + 5} \right)}^{3}}{\left(3 x + 5\right)^{2}}$$
The third derivative [src]
       2/  _________\ /3      2/  _________\        /  _________\\
135*log \\/ 5 + 3*x /*|- + log \\/ 5 + 3*x / - 3*log\\/ 5 + 3*x /|
                      \2                                         /
------------------------------------------------------------------
                                     3                            
                            (5 + 3*x)                             
$$\frac{135 \left(\log{\left(\sqrt{3 x + 5} \right)}^{2} - 3 \log{\left(\sqrt{3 x + 5} \right)} + \frac{3}{2}\right) \log{\left(\sqrt{3 x + 5} \right)}^{2}}{\left(3 x + 5\right)^{3}}$$
The graph
Derivative of y=ln^5(sqrt(3x+5))