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y=ln^2(4x+5)

Derivative of y=ln^2(4x+5)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   2         
log (4*x + 5)
$$\log{\left(4 x + 5 \right)}^{2}$$
log(4*x + 5)^2
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. 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:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
8*log(4*x + 5)
--------------
   4*x + 5    
$$\frac{8 \log{\left(4 x + 5 \right)}}{4 x + 5}$$
The second derivative [src]
32*(1 - log(5 + 4*x))
---------------------
               2     
      (5 + 4*x)      
$$\frac{32 \left(1 - \log{\left(4 x + 5 \right)}\right)}{\left(4 x + 5\right)^{2}}$$
The third derivative [src]
128*(-3 + 2*log(5 + 4*x))
-------------------------
                 3       
        (5 + 4*x)        
$$\frac{128 \left(2 \log{\left(4 x + 5 \right)} - 3\right)}{\left(4 x + 5\right)^{3}}$$
The graph
Derivative of y=ln^2(4x+5)