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y=(sqrt3x+1)-sin2x

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Identical expressions
y=(sqrt3x+ one)-sin2x
y equally ( square root of 3x plus 1) minus sinus of 2x
y equally ( square root of 3x plus one) minus sinus of 2x
y=(√3x+1)-sin2x
y=sqrt3x+1-sin2x
Similar expressions
y=(sqrt3x-1)-sin2x
y=(sqrt3x+1)+sin2x

The derivative of the function/ y=(sqrt3x+1)-sin2x

Derivative of y=(sqrt3x+1)-sin2x

The solution

You have entered [src]

  _____               
\/ 3*x  + 1 - sin(2*x)

$$\left(\sqrt{3 x} + 1\right) - \sin{\left(2 x \right)}$$

sqrt(3*x) + 1 - sin(2*x)

Detail solution

Differentiate $\left(\sqrt{3 x} + 1\right) - \sin{\left(2 x \right)}$ term by term:
1. Differentiate $\sqrt{3 x} + 1$ term by term:
  1. Let $u = 3 x$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
    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: $3$
    The result of the chain rule is:
    
    $\frac{\sqrt{3}}{2 \sqrt{x}}$
  4. The derivative of the constant $1$ is zero.
  The result is: $\frac{\sqrt{3}}{2 \sqrt{x}}$
2. 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: $- 2 \cos{\left(2 x \right)}$
The result is: $- 2 \cos{\left(2 x \right)} + \frac{\sqrt{3}}{2 \sqrt{x}}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of y=(sqrt3x+1)-sin2x