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y=log(3)(4x+5)/(2ctgsqrtx)

Derivative of y=log(3)(4x+5)/(2ctgsqrtx)

Function f() - derivative -N order at the point
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The solution

You have entered [src]
log(3)*(4*x + 5)
----------------
       /  ___\  
  2*cot\\/ x /  
$$\frac{\left(4 x + 5\right) \log{\left(3 \right)}}{2 \cot{\left(\sqrt{x} \right)}}$$
(log(3)*(4*x + 5))/((2*cot(sqrt(x))))
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Differentiate term by term:

        1. The derivative of the constant is zero.

        2. 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:

        The result is:

      So, the result is:

    To find :

    1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. There are multiple ways to do this derivative.

        Method #1

        1. Rewrite the function to be differentiated:

        2. Let .

        3. Apply the power rule: goes to

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

          1. Rewrite the function to be differentiated:

          2. Apply the quotient rule, which is:

            and .

            To find :

            1. Let .

            2. The derivative of sine is cosine:

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

              1. Apply the power rule: goes to

              The result of the chain rule is:

            To find :

            1. Let .

            2. The derivative of cosine is negative sine:

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

              1. Apply the power rule: goes to

              The result of the chain rule is:

            Now plug in to the quotient rule:

          The result of the chain rule is:

        Method #2

        1. Rewrite the function to be differentiated:

        2. Apply the quotient rule, which is:

          and .

          To find :

          1. Let .

          2. The derivative of cosine is negative sine:

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

            1. Apply the power rule: goes to

            The result of the chain rule is:

          To find :

          1. Let .

          2. The derivative of sine is cosine:

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

            1. Apply the power rule: goes to

            The result of the chain rule is:

          Now plug in to the quotient rule:

      So, the result is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
                        /        2/  ___\\                 
       1                \-1 - cot \\/ x //*(4*x + 5)*log(3)
4*------------*log(3) - -----------------------------------
       /  ___\                      ___    2/  ___\        
  2*cot\\/ x /                  4*\/ x *cot \\/ x /        
$$4 \log{\left(3 \right)} \frac{1}{2 \cot{\left(\sqrt{x} \right)}} - \frac{\left(4 x + 5\right) \left(- \cot^{2}{\left(\sqrt{x} \right)} - 1\right) \log{\left(3 \right)}}{4 \sqrt{x} \cot^{2}{\left(\sqrt{x} \right)}}$$
The second derivative [src]
                  /                             /                        /       2/  ___\\\\       
                  |                             |2          1          2*\1 + cot \\/ x //||       
                  |                   (5 + 4*x)*|- + --------------- - -------------------||       
                  |                             |x    3/2    /  ___\           2/  ___\   ||       
/       2/  ___\\ |       2                     \    x   *cot\\/ x /      x*cot \\/ x /   /|       
\1 + cot \\/ x //*|---------------- - -----------------------------------------------------|*log(3)
                  |  ___    /  ___\                             8                          |       
                  \\/ x *cot\\/ x /                                                        /       
---------------------------------------------------------------------------------------------------
                                                /  ___\                                            
                                             cot\\/ x /                                            
$$\frac{\left(- \frac{\left(4 x + 5\right) \left(- \frac{2 \left(\cot^{2}{\left(\sqrt{x} \right)} + 1\right)}{x \cot^{2}{\left(\sqrt{x} \right)}} + \frac{2}{x} + \frac{1}{x^{\frac{3}{2}} \cot{\left(\sqrt{x} \right)}}\right)}{8} + \frac{2}{\sqrt{x} \cot{\left(\sqrt{x} \right)}}\right) \left(\cot^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(3 \right)}}{\cot{\left(\sqrt{x} \right)}}$$
The third derivative [src]
                  /                                                                                                                             /                        /       2/  ___\\\\       
                  |                                                                                                                             |2          1          2*\1 + cot \\/ x //||       
                  |          /                                                                                                          2\   24*|- + --------------- - -------------------||       
                  |          |                                             /       2/  ___\\     /       2/  ___\\     /       2/  ___\\ |      |x    3/2    /  ___\           2/  ___\   ||       
/       2/  ___\\ |          | 4            3                 6         10*\1 + cot \\/ x //   6*\1 + cot \\/ x //   6*\1 + cot \\/ x // |      \    x   *cot\\/ x /      x*cot \\/ x /   /|       
\1 + cot \\/ x //*|(5 + 4*x)*|---- + ---------------- + ------------- - -------------------- - ------------------- + --------------------| - ----------------------------------------------|*log(3)
                  |          | 3/2    5/2    2/  ___\    2    /  ___\      3/2    2/  ___\         2    3/  ___\        3/2    4/  ___\  |                        /  ___\                  |       
                  \          \x      x   *cot \\/ x /   x *cot\\/ x /     x   *cot \\/ x /        x *cot \\/ x /       x   *cot \\/ x /  /                     cot\\/ x /                  /       
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                 16                                                                                                
$$\frac{\left(\left(4 x + 5\right) \left(- \frac{6 \left(\cot^{2}{\left(\sqrt{x} \right)} + 1\right)}{x^{2} \cot^{3}{\left(\sqrt{x} \right)}} + \frac{6}{x^{2} \cot{\left(\sqrt{x} \right)}} + \frac{6 \left(\cot^{2}{\left(\sqrt{x} \right)} + 1\right)^{2}}{x^{\frac{3}{2}} \cot^{4}{\left(\sqrt{x} \right)}} - \frac{10 \left(\cot^{2}{\left(\sqrt{x} \right)} + 1\right)}{x^{\frac{3}{2}} \cot^{2}{\left(\sqrt{x} \right)}} + \frac{4}{x^{\frac{3}{2}}} + \frac{3}{x^{\frac{5}{2}} \cot^{2}{\left(\sqrt{x} \right)}}\right) - \frac{24 \left(- \frac{2 \left(\cot^{2}{\left(\sqrt{x} \right)} + 1\right)}{x \cot^{2}{\left(\sqrt{x} \right)}} + \frac{2}{x} + \frac{1}{x^{\frac{3}{2}} \cot{\left(\sqrt{x} \right)}}\right)}{\cot{\left(\sqrt{x} \right)}}\right) \left(\cot^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(3 \right)}}{16}$$
The graph
Derivative of y=log(3)(4x+5)/(2ctgsqrtx)