Derivative of y(x)=sinx√cos2x

The solution

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         __________
sin(x)*\/ cos(2*x)

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

d /         __________\
--\sin(x)*\/ cos(2*x) /
dx

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

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
$g{\left(x \right)} = \sqrt{\cos{\left(2 x \right)}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \cos{\left(2 x \right)}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(2 x \right)}$ :
  1. Let $u = 2 x$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  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:
    
    $- 2 \sin{\left(2 x \right)}$
  The result of the chain rule is:
  
  $- \frac{\sin{\left(2 x \right)}}{\sqrt{\cos{\left(2 x \right)}}}$
The result is: $- \frac{\sin{\left(x \right)} \sin{\left(2 x \right)}}{\sqrt{\cos{\left(2 x \right)}}} + \cos{\left(x \right)} \sqrt{\cos{\left(2 x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y(x)=sinx√cos2x

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The solution