Mister Exam

Derivative of y=x×tg3x+2^x

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
              x
x*tan(3*x) + 2 
$$2^{x} + x \tan{\left(3 x \right)}$$
x*tan(3*x) + 2^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:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
  /         2     \    x                  
x*\3 + 3*tan (3*x)/ + 2 *log(2) + tan(3*x)
$$2^{x} \log{\left(2 \right)} + x \left(3 \tan^{2}{\left(3 x \right)} + 3\right) + \tan{\left(3 x \right)}$$
The second derivative [src]
         2         x    2           /       2     \         
6 + 6*tan (3*x) + 2 *log (2) + 18*x*\1 + tan (3*x)/*tan(3*x)
$$2^{x} \log{\left(2 \right)}^{2} + 18 x \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} + 6 \tan^{2}{\left(3 x \right)} + 6$$
The third derivative [src]
                                 2                                                                
 x    3           /       2     \       /       2     \                     2      /       2     \
2 *log (2) + 54*x*\1 + tan (3*x)/  + 54*\1 + tan (3*x)/*tan(3*x) + 108*x*tan (3*x)*\1 + tan (3*x)/
$$2^{x} \log{\left(2 \right)}^{3} + 54 x \left(\tan^{2}{\left(3 x \right)} + 1\right)^{2} + 108 x \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan^{2}{\left(3 x \right)} + 54 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)}$$
The graph
Derivative of y=x×tg3x+2^x