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y=(sqrt(x-2))*sin(3x-2)

Derivative of y=(sqrt(x-2))*sin(3x-2)

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

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The solution

You have entered [src]
  _______             
\/ x - 2 *sin(3*x - 2)
$$\sqrt{x - 2} \sin{\left(3 x - 2 \right)}$$
sqrt(x - 2)*sin(3*x - 2)
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Let .

    2. Apply the power rule: goes to

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

      1. Differentiate term by term:

        1. Apply the power rule: goes to

        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:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
sin(3*x - 2)       _______             
------------ + 3*\/ x - 2 *cos(3*x - 2)
    _______                            
2*\/ x - 2                             
$$3 \sqrt{x - 2} \cos{\left(3 x - 2 \right)} + \frac{\sin{\left(3 x - 2 \right)}}{2 \sqrt{x - 2}}$$
The second derivative [src]
      ________                 3*cos(-2 + 3*x)   sin(-2 + 3*x)
- 9*\/ -2 + x *sin(-2 + 3*x) + --------------- - -------------
                                    ________               3/2
                                  \/ -2 + x      4*(-2 + x)   
$$- 9 \sqrt{x - 2} \sin{\left(3 x - 2 \right)} + \frac{3 \cos{\left(3 x - 2 \right)}}{\sqrt{x - 2}} - \frac{\sin{\left(3 x - 2 \right)}}{4 \left(x - 2\right)^{\frac{3}{2}}}$$
The third derivative [src]
  /      ________                 9*sin(-2 + 3*x)   3*cos(-2 + 3*x)   sin(-2 + 3*x)\
3*|- 9*\/ -2 + x *cos(-2 + 3*x) - --------------- - --------------- + -------------|
  |                                     ________               3/2              5/2|
  \                                 2*\/ -2 + x      4*(-2 + x)       8*(-2 + x)   /
$$3 \left(- 9 \sqrt{x - 2} \cos{\left(3 x - 2 \right)} - \frac{9 \sin{\left(3 x - 2 \right)}}{2 \sqrt{x - 2}} - \frac{3 \cos{\left(3 x - 2 \right)}}{4 \left(x - 2\right)^{\frac{3}{2}}} + \frac{\sin{\left(3 x - 2 \right)}}{8 \left(x - 2\right)^{\frac{5}{2}}}\right)$$
The graph
Derivative of y=(sqrt(x-2))*sin(3x-2)