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Identical expressions
sqrt(x- two)*(sin(3x- two))
square root of (x minus 2) multiply by ( sinus of (3x minus 2))
square root of (x minus two) multiply by ( sinus of (3x minus two))
√(x-2)*(sin(3x-2))
sqrt(x-2)(sin(3x-2))
sqrtx-2sin3x-2
Similar expressions
y=sqrt(x-2)*sin(3x-2)
sqrt(x+2)*(sin(3x-2))
sqrt(x-2)*(sin(3x+2))

The derivative of the function/ sqrt(x-2)*(sin(3x-2))

Derivative of sqrt(x-2)*(sin(3x-2))

The solution

You have entered [src]

  _______             
\/ x - 2 *sin(3*x - 2)

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

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

Detail solution

Apply the product rule:

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

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

$\frac{\left(6 x - 12\right) \cos{\left(3 x - 2 \right)} + \sin{\left(3 x - 2 \right)}}{2 \sqrt{x - 2}}$

The answer is:

$\frac{\left(6 x - 12\right) \cos{\left(3 x - 2 \right)} + \sin{\left(3 x - 2 \right)}}{2 \sqrt{x - 2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

      ________                 3*cos(-2 + 3*x)   sin(-2 + 3*x)
- 9*\/ -2 + x *sin(-2 + 3*x) + --------------- - -------------
                                    ________               3/2
                                  \/ -2 + x      4*(-2 + x)

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

Simplify

The third derivative [src]

  /      ________                 9*sin(-2 + 3*x)   3*cos(-2 + 3*x)   sin(-2 + 3*x)\
3*|- 9*\/ -2 + x *cos(-2 + 3*x) - --------------- - --------------- + -------------|
  |                                     ________               3/2              5/2|
  \                                 2*\/ -2 + x      4*(-2 + x)       8*(-2 + x)   /

$$3 \left(- 9 \sqrt{x - 2} \cos{\left(3 x - 2 \right)} - \frac{9 \sin{\left(3 x - 2 \right)}}{2 \sqrt{x - 2}} - \frac{3 \cos{\left(3 x - 2 \right)}}{4 \left(x - 2\right)^{\frac{3}{2}}} + \frac{\sin{\left(3 x - 2 \right)}}{8 \left(x - 2\right)^{\frac{5}{2}}}\right)$$

Simplify

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Derivative of sqrt(x-2)*(sin(3x-2))

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The solution