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y=xtan(2√x)+7sin(1/x)

Derivative of y=xtan(2√x)+7sin(1/x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
     /    ___\        /1\
x*tan\2*\/ x / + 7*sin|-|
                      \x/
$$x \tan{\left(2 \sqrt{x} \right)} + 7 \sin{\left(\frac{1}{x} \right)}$$
x*tan(2*sqrt(x)) + 7*sin(1/x)
Detail solution
  1. Differentiate term by term:

    1. Apply the product rule:

      ; to find :

      1. Apply the power rule: goes to

      ; to find :

      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. 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 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. 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 of the chain rule is:

        Now plug in to the quotient rule:

      The result is:

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

      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:

      So, the result is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
                                 /1\               
                            7*cos|-|               
  ___ /       2/    ___\\        \x/      /    ___\
\/ x *\1 + tan \2*\/ x // - -------- + tan\2*\/ x /
                                2                  
                               x                   
$$\sqrt{x} \left(\tan^{2}{\left(2 \sqrt{x} \right)} + 1\right) + \tan{\left(2 \sqrt{x} \right)} - \frac{7 \cos{\left(\frac{1}{x} \right)}}{x^{2}}$$
The second derivative [src]
       /1\                                              /1\                        
  7*sin|-|                                        14*cos|-|     /       2/    ___\\
       \x/     /       2/    ___\\    /    ___\         \x/   3*\1 + tan \2*\/ x //
- -------- + 2*\1 + tan \2*\/ x //*tan\2*\/ x / + --------- + ---------------------
      4                                                3                 ___       
     x                                                x              2*\/ x        
$$2 \left(\tan^{2}{\left(2 \sqrt{x} \right)} + 1\right) \tan{\left(2 \sqrt{x} \right)} + \frac{14 \cos{\left(\frac{1}{x} \right)}}{x^{3}} - \frac{7 \sin{\left(\frac{1}{x} \right)}}{x^{4}} + \frac{3 \left(\tan^{2}{\left(2 \sqrt{x} \right)} + 1\right)}{2 \sqrt{x}}$$
The third derivative [src]
        /1\                        2        /1\         /1\                                                                                                   
  42*cos|-|     /       2/    ___\\    7*cos|-|   42*sin|-|     /       2/    ___\\     /       2/    ___\\    /    ___\        2/    ___\ /       2/    ___\\
        \x/   2*\1 + tan \2*\/ x //         \x/         \x/   3*\1 + tan \2*\/ x //   3*\1 + tan \2*\/ x //*tan\2*\/ x /   4*tan \2*\/ x /*\1 + tan \2*\/ x //
- --------- + ---------------------- + -------- + --------- - --------------------- + ---------------------------------- + -----------------------------------
       4                ___                6           5                 3/2                          x                                     ___               
      x               \/ x                x           x               4*x                                                                 \/ x                
$$\frac{3 \left(\tan^{2}{\left(2 \sqrt{x} \right)} + 1\right) \tan{\left(2 \sqrt{x} \right)}}{x} - \frac{42 \cos{\left(\frac{1}{x} \right)}}{x^{4}} + \frac{42 \sin{\left(\frac{1}{x} \right)}}{x^{5}} + \frac{7 \cos{\left(\frac{1}{x} \right)}}{x^{6}} + \frac{2 \left(\tan^{2}{\left(2 \sqrt{x} \right)} + 1\right)^{2}}{\sqrt{x}} + \frac{4 \left(\tan^{2}{\left(2 \sqrt{x} \right)} + 1\right) \tan^{2}{\left(2 \sqrt{x} \right)}}{\sqrt{x}} - \frac{3 \left(\tan^{2}{\left(2 \sqrt{x} \right)} + 1\right)}{4 x^{\frac{3}{2}}}$$
The graph
Derivative of y=xtan(2√x)+7sin(1/x)