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Derivative of 2*tan(3x)+3*tan(2x)

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The solution

You have entered [src]
2*tan(3*x) + 3*tan(2*x)
$$3 \tan{\left(2 x \right)} + 2 \tan{\left(3 x \right)}$$
2*tan(3*x) + 3*tan(2*x)
Detail solution
  1. Differentiate term by term:

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

      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:

      So, the result is:

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

      1. Let .

      2. 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:

      So, the result is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
          2             2     
12 + 6*tan (2*x) + 6*tan (3*x)
$$6 \tan^{2}{\left(2 x \right)} + 6 \tan^{2}{\left(3 x \right)} + 12$$
The second derivative [src]
   /  /       2     \              /       2     \         \
12*\2*\1 + tan (2*x)/*tan(2*x) + 3*\1 + tan (3*x)/*tan(3*x)/
$$12 \left(2 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)} + 3 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)}\right)$$
3-я производная [src]
   /                 2                    2                                                             \
   |  /       2     \      /       2     \         2      /       2     \         2      /       2     \|
12*\4*\1 + tan (2*x)/  + 9*\1 + tan (3*x)/  + 8*tan (2*x)*\1 + tan (2*x)/ + 18*tan (3*x)*\1 + tan (3*x)//
$$12 \left(4 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2} + 8 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan^{2}{\left(2 x \right)} + 9 \left(\tan^{2}{\left(3 x \right)} + 1\right)^{2} + 18 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan^{2}{\left(3 x \right)}\right)$$
The third derivative [src]
   /                 2                    2                                                             \
   |  /       2     \      /       2     \         2      /       2     \         2      /       2     \|
12*\4*\1 + tan (2*x)/  + 9*\1 + tan (3*x)/  + 8*tan (2*x)*\1 + tan (2*x)/ + 18*tan (3*x)*\1 + tan (3*x)//
$$12 \left(4 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2} + 8 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan^{2}{\left(2 x \right)} + 9 \left(\tan^{2}{\left(3 x \right)} + 1\right)^{2} + 18 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan^{2}{\left(3 x \right)}\right)$$