Derivative of sqrt(x)*(cosx)

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  ___       
\/ x *cos(x)

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

sqrt(x)*cos(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 $\frac{1}{2 \sqrt{x}}$
$g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
The result is: $- \sqrt{x} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{2 \sqrt{x}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  ___          3*cos(x)   3*sin(x)   3*cos(x)
\/ x *sin(x) - -------- + -------- + --------
                   ___        3/2        5/2 
               2*\/ x      4*x        8*x

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

Simplify

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Derivative of sqrt(x)*(cosx)

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The solution