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The derivative of the function/ 1/2sin2x+√2x

Derivative of 1/2sin2x+√2x

The solution

You have entered [src]

sin(2*x)     _____
-------- + \/ 2*x 
   2

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

sin(2*x)/2 + sqrt(2*x)

Detail solution

Differentiate $\sqrt{2 x} + \frac{\sin{\left(2 x \right)}}{2}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  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:
    
    $2 \cos{\left(2 x \right)}$
  So, the result is: $\cos{\left(2 x \right)}$
2. Let $u = 2 x$ .
3. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
4. 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:
  
  $\frac{\sqrt{2}}{2 \sqrt{x}}$
The result is: $\cos{\left(2 x \right)} + \frac{\sqrt{2}}{2 \sqrt{x}}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify