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y=sqrt(x)sin(x+6)

Derivative of y=sqrt(x)sin(x+6)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
  ___           
\/ x *sin(x + 6)
$$\sqrt{x} \sin{\left(x + 6 \right)}$$
sqrt(x)*sin(x + 6)
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Apply the power rule: goes to

    ; 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. Apply the power rule: goes to

        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(x + 6)
\/ x *cos(x + 6) + ----------
                        ___  
                    2*\/ x   
$$\sqrt{x} \cos{\left(x + 6 \right)} + \frac{\sin{\left(x + 6 \right)}}{2 \sqrt{x}}$$
The second derivative [src]
cos(6 + x)     ___              sin(6 + x)
---------- - \/ x *sin(6 + x) - ----------
    ___                              3/2  
  \/ x                            4*x     
$$- \sqrt{x} \sin{\left(x + 6 \right)} + \frac{\cos{\left(x + 6 \right)}}{\sqrt{x}} - \frac{\sin{\left(x + 6 \right)}}{4 x^{\frac{3}{2}}}$$
The third derivative [src]
    ___              3*sin(6 + x)   3*cos(6 + x)   3*sin(6 + x)
- \/ x *cos(6 + x) - ------------ - ------------ + ------------
                           ___            3/2            5/2   
                       2*\/ x          4*x            8*x      
$$- \sqrt{x} \cos{\left(x + 6 \right)} - \frac{3 \sin{\left(x + 6 \right)}}{2 \sqrt{x}} - \frac{3 \cos{\left(x + 6 \right)}}{4 x^{\frac{3}{2}}} + \frac{3 \sin{\left(x + 6 \right)}}{8 x^{\frac{5}{2}}}$$
The graph
Derivative of y=sqrt(x)sin(x+6)