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Derivative of exp(sin(2y))-3*exp(cos(2y))

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

You have entered [src]
 sin(2*y)      cos(2*y)
e         - 3*e        
$$e^{\sin{\left(2 y \right)}} - 3 e^{\cos{\left(2 y \right)}}$$
exp(sin(2*y)) - 3*exp(cos(2*y))
Detail solution
  1. Differentiate term by term:

    1. Let .

    2. The derivative of is itself.

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

      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:

      The result of the chain rule is:

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

      1. Let .

      2. The derivative of is itself.

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

        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:

        The result of the chain rule is:

      So, the result is:

    The result is:


The answer is:

The graph
The first derivative [src]
            sin(2*y)      cos(2*y)         
2*cos(2*y)*e         + 6*e        *sin(2*y)
$$2 e^{\sin{\left(2 y \right)}} \cos{\left(2 y \right)} + 6 e^{\cos{\left(2 y \right)}} \sin{\left(2 y \right)}$$
The second derivative [src]
  /   2       sin(2*y)    sin(2*y)                 2       cos(2*y)               cos(2*y)\
4*\cos (2*y)*e         - e        *sin(2*y) - 3*sin (2*y)*e         + 3*cos(2*y)*e        /
$$4 \left(- e^{\sin{\left(2 y \right)}} \sin{\left(2 y \right)} + e^{\sin{\left(2 y \right)}} \cos^{2}{\left(2 y \right)} - 3 e^{\cos{\left(2 y \right)}} \sin^{2}{\left(2 y \right)} + 3 e^{\cos{\left(2 y \right)}} \cos{\left(2 y \right)}\right)$$
The third derivative [src]
  /   3       sin(2*y)             sin(2*y)      cos(2*y)                 3       cos(2*y)               cos(2*y)                        sin(2*y)         \
8*\cos (2*y)*e         - cos(2*y)*e         - 3*e        *sin(2*y) + 3*sin (2*y)*e         - 9*cos(2*y)*e        *sin(2*y) - 3*cos(2*y)*e        *sin(2*y)/
$$8 \left(- 3 e^{\sin{\left(2 y \right)}} \sin{\left(2 y \right)} \cos{\left(2 y \right)} + e^{\sin{\left(2 y \right)}} \cos^{3}{\left(2 y \right)} - e^{\sin{\left(2 y \right)}} \cos{\left(2 y \right)} + 3 e^{\cos{\left(2 y \right)}} \sin^{3}{\left(2 y \right)} - 9 e^{\cos{\left(2 y \right)}} \sin{\left(2 y \right)} \cos{\left(2 y \right)} - 3 e^{\cos{\left(2 y \right)}} \sin{\left(2 y \right)}\right)$$