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The derivative of the function/ sin(2*x)/sqrt(x)

Derivative of sin(2*x)/sqrt(x)

The solution

You have entered [src]

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

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

d /sin(2*x)\
--|--------|
dx|   ___  |
  \ \/ x   /

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

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(2 x \right)}$ and $g{\left(x \right)} = \sqrt{x}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 x$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $2$
  The result of the chain rule is:
  
  $2 \cos{\left(2 x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
Now plug in to the quotient rule:

$\frac{2 \sqrt{x} \cos{\left(2 x \right)} - \frac{\sin{\left(2 x \right)}}{2 \sqrt{x}}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

              6*sin(2*x)   15*sin(2*x)   9*cos(2*x)
-8*cos(2*x) + ---------- - ----------- + ----------
                  x               3            2   
                               8*x          2*x    
---------------------------------------------------
                         ___                       
                       \/ x

$$\frac{- 8 \cos{\left(2 x \right)} + \frac{6 \sin{\left(2 x \right)}}{x} + \frac{9 \cos{\left(2 x \right)}}{2 x^{2}} - \frac{15 \sin{\left(2 x \right)}}{8 x^{3}}}{\sqrt{x}}$$

Simplify

The graph

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Derivative of sin(2*x)/sqrt(x)

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The solution