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sqrt(3*x+1)*sin(2*x)

Derivative of sqrt(3*x+1)*sin(2*x)

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

from to

Piecewise:

The solution

You have entered [src]
  _________         
\/ 3*x + 1 *sin(2*x)
$$\sqrt{3 x + 1} \sin{\left(2 x \right)}$$
sqrt(3*x + 1)*sin(2*x)
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. 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. 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 is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
    _________              3*sin(2*x) 
2*\/ 3*x + 1 *cos(2*x) + -------------
                             _________
                         2*\/ 3*x + 1 
$$2 \sqrt{3 x + 1} \cos{\left(2 x \right)} + \frac{3 \sin{\left(2 x \right)}}{2 \sqrt{3 x + 1}}$$
The second derivative [src]
      _________             6*cos(2*x)     9*sin(2*x)  
- 4*\/ 1 + 3*x *sin(2*x) + ----------- - --------------
                             _________              3/2
                           \/ 1 + 3*x    4*(1 + 3*x)   
$$- 4 \sqrt{3 x + 1} \sin{\left(2 x \right)} + \frac{6 \cos{\left(2 x \right)}}{\sqrt{3 x + 1}} - \frac{9 \sin{\left(2 x \right)}}{4 \left(3 x + 1\right)^{\frac{3}{2}}}$$
The third derivative [src]
  18*sin(2*x)       _________             27*cos(2*x)      81*sin(2*x)  
- ----------- - 8*\/ 1 + 3*x *cos(2*x) - -------------- + --------------
    _________                                       3/2              5/2
  \/ 1 + 3*x                             2*(1 + 3*x)      8*(1 + 3*x)   
$$- 8 \sqrt{3 x + 1} \cos{\left(2 x \right)} - \frac{18 \sin{\left(2 x \right)}}{\sqrt{3 x + 1}} - \frac{27 \cos{\left(2 x \right)}}{2 \left(3 x + 1\right)^{\frac{3}{2}}} + \frac{81 \sin{\left(2 x \right)}}{8 \left(3 x + 1\right)^{\frac{5}{2}}}$$
The graph
Derivative of sqrt(3*x+1)*sin(2*x)