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y=ctg(6x^2-9)

Derivative of y=ctg(6x^2-9)

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

You have entered [src]
   /   2    \
cot\6*x  - 9/
$$\cot{\left(6 x^{2} - 9 \right)}$$
cot(6*x^2 - 9)
Detail solution
  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. 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:

        To find :

        1. Let .

        2. The derivative of cosine is negative sine:

        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:

        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. 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:

      To find :

      1. Let .

      2. The derivative of sine is cosine:

      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:

      Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
     /        2/   2    \\
12*x*\-1 - cot \6*x  - 9//
$$12 x \left(- \cot^{2}{\left(6 x^{2} - 9 \right)} - 1\right)$$
The second derivative [src]
   /        2/  /        2\\       2 /       2/  /        2\\\    /  /        2\\\
12*\-1 - cot \3*\-3 + 2*x // + 24*x *\1 + cot \3*\-3 + 2*x ///*cot\3*\-3 + 2*x ///
$$12 \left(24 x^{2} \left(\cot^{2}{\left(3 \left(2 x^{2} - 3\right) \right)} + 1\right) \cot{\left(3 \left(2 x^{2} - 3\right) \right)} - \cot^{2}{\left(3 \left(2 x^{2} - 3\right) \right)} - 1\right)$$
The third derivative [src]
      /       2/  /        2\\\ /     2    2/  /        2\\      2 /       2/  /        2\\\      /  /        2\\\
864*x*\1 + cot \3*\-3 + 2*x ///*\- 8*x *cot \3*\-3 + 2*x // - 4*x *\1 + cot \3*\-3 + 2*x /// + cot\3*\-3 + 2*x ///
$$864 x \left(\cot^{2}{\left(3 \left(2 x^{2} - 3\right) \right)} + 1\right) \left(- 4 x^{2} \left(\cot^{2}{\left(3 \left(2 x^{2} - 3\right) \right)} + 1\right) - 8 x^{2} \cot^{2}{\left(3 \left(2 x^{2} - 3\right) \right)} + \cot{\left(3 \left(2 x^{2} - 3\right) \right)}\right)$$
The graph
Derivative of y=ctg(6x^2-9)