Derivative of sqrt(x)*sin(x)

The solution

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  ___       
\/ x *sin(x)

$$\sqrt{x} \sin{\left(x \right)}$$

sqrt(x)*sin(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sqrt{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
$g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
The result is: $\sqrt{x} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{2 \sqrt{x}}$
Now simplify:

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

The answer is:

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

The first derivative [src]

  ___           sin(x)
\/ x *cos(x) + -------
                   ___
               2*\/ x

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

Simplify

The second derivative [src]

cos(x)     ___          sin(x)
------ - \/ x *sin(x) - ------
  ___                      3/2
\/ x                    4*x

$$- \sqrt{x} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{\sqrt{x}} - \frac{\sin{\left(x \right)}}{4 x^{\frac{3}{2}}}$$

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

$$- \sqrt{x} \cos{\left(x \right)} - \frac{3 \sin{\left(x \right)}}{2 \sqrt{x}} - \frac{3 \cos{\left(x \right)}}{4 x^{\frac{3}{2}}} + \frac{3 \sin{\left(x \right)}}{8 x^{\frac{5}{2}}}$$

Simplify

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Derivative of sqrt(x)*sin(x)

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