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sqrt(x)*sin(x)
The derivative of the function/ sqrt(x)*sin(x)

Derivative of sqrt(x)*sin(x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  ___       
\/ x *sin(x)
xsin⁡(x)\sqrt{x} \sin{\left(x \right)} 
sqrt(x)*sin(x)
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=xf{\left(x \right)} = \sqrt{x}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the power rule: x\sqrt{x} goes to 12x\frac{1}{2 \sqrt{x}}

    g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. The derivative of sine is cosine:

      ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

    The result is: xcos⁡(x)+sin⁡(x)2x\sqrt{x} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{2 \sqrt{x}}

  2. Now simplify:

    xcos⁡(x)+sin⁡(x)2x\frac{x \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{2}}{\sqrt{x}}

The answer is:

xcos⁡(x)+sin⁡(x)2x\frac{x \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{2}}{\sqrt{x}}

The first derivative [src] 
  ___           sin(x)
\/ x *cos(x) + -------
                   ___
               2*\/ x
xcos⁡(x)+sin⁡(x)2x\sqrt{x} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{2 \sqrt{x}}
Simplify
The second derivative [src] 
cos(x)     ___          sin(x)
------ - \/ x *sin(x) - ------
  ___                      3/2
\/ x                    4*x
−xsin⁡(x)+cos⁡(x)x−sin⁡(x)4x32- \sqrt{x} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{\sqrt{x}} - \frac{\sin{\left(x \right)}}{4 x^{\frac{3}{2}}}
Simplify
The third derivative [src] 
    ___          3*sin(x)   3*cos(x)   3*sin(x)
- \/ x *cos(x) - -------- - -------- + --------
                     ___        3/2        5/2 
                 2*\/ x      4*x        8*x
−xcos⁡(x)−3sin⁡(x)2x−3cos⁡(x)4x32+3sin⁡(x)8x52- \sqrt{x} \cos{\left(x \right)} - \frac{3 \sin{\left(x \right)}}{2 \sqrt{x}} - \frac{3 \cos{\left(x \right)}}{4 x^{\frac{3}{2}}} + \frac{3 \sin{\left(x \right)}}{8 x^{\frac{5}{2}}}
Simplify
The graph
Derivative of sqrt(x)*sin(x)
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