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Derivative of:
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Identical expressions
y=sqrt(3x+ one)*sin(2x)
y equally square root of (3x plus 1) multiply by sinus of (2x)
y equally square root of (3x plus one) multiply by sinus of (2x)
y=√(3x+1)*sin(2x)
y=sqrt(3x+1)sin(2x)
y=sqrt3x+1sin2x
Similar expressions
y=sqrt(3x-1)*sin(2x)
sqrt(3*x+1)*sin(2*x)

The derivative of the function/ y=sqrt(3x+1)*sin(2x)

Derivative of y=sqrt(3x+1)*sin(2x)

The solution

You have entered [src]

  _________         
\/ 3*x + 1 *sin(2*x)

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

d /  _________         \
--\\/ 3*x + 1 *sin(2*x)/
dx

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

Transform

Detail solution

Apply the product rule:

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

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

$\frac{\left(12 x + 4\right) \cos{\left(2 x \right)} + 3 \sin{\left(2 x \right)}}{2 \sqrt{3 x + 1}}$

The answer is:

$\frac{\left(12 x + 4\right) \cos{\left(2 x \right)} + 3 \sin{\left(2 x \right)}}{2 \sqrt{3 x + 1}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

      _________             6*cos(2*x)     9*sin(2*x)  
- 4*\/ 1 + 3*x *sin(2*x) + ----------- - --------------
                             _________              3/2
                           \/ 1 + 3*x    4*(1 + 3*x)

$$- 4 \sqrt{3 x + 1} \sin{\left(2 x \right)} + \frac{6 \cos{\left(2 x \right)}}{\sqrt{3 x + 1}} - \frac{9 \sin{\left(2 x \right)}}{4 \left(3 x + 1\right)^{\frac{3}{2}}}$$

Simplify

The third derivative [src]

  18*sin(2*x)       _________             27*cos(2*x)      81*sin(2*x)  
- ----------- - 8*\/ 1 + 3*x *cos(2*x) - -------------- + --------------
    _________                                       3/2              5/2
  \/ 1 + 3*x                             2*(1 + 3*x)      8*(1 + 3*x)

$$- 8 \sqrt{3 x + 1} \cos{\left(2 x \right)} - \frac{18 \sin{\left(2 x \right)}}{\sqrt{3 x + 1}} - \frac{27 \cos{\left(2 x \right)}}{2 \left(3 x + 1\right)^{\frac{3}{2}}} + \frac{81 \sin{\left(2 x \right)}}{8 \left(3 x + 1\right)^{\frac{5}{2}}}$$

Simplify

The graph

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Derivative of y=sqrt(3x+1)*sin(2x)

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The solution