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Similar expressions
sen(2x)ln((x+1)^(1/2))

The derivative of the function/ sen(2x)ln((x-1)^(1/2))

Derivative of sen(2x)ln((x-1)^(1/2))

The solution

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            /  _______\
sin(2*x)*log\\/ x - 1 /

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

sin(2*x)*log(sqrt(x - 1))

Detail solution

Apply the product rule:

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

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

$\frac{\left(x - 1\right) \log{\left(x - 1 \right)} \cos{\left(2 x \right)} + \frac{\sin{\left(2 x \right)}}{2}}{x - 1}$

The answer is:

$\frac{\left(x - 1\right) \log{\left(x - 1 \right)} \cos{\left(2 x \right)} + \frac{\sin{\left(2 x \right)}}{2}}{x - 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 sin(2*x)                 /  _______\
--------- + 2*cos(2*x)*log\\/ x - 1 /
2*(x - 1)

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

Simplify

The second derivative [src]

       /  ________\            2*cos(2*x)     sin(2*x) 
- 4*log\\/ -1 + x /*sin(2*x) + ---------- - -----------
                                 -1 + x               2
                                            2*(-1 + x)

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

Simplify

The third derivative [src]

 sin(2*x)                 /  ________\   6*sin(2*x)   3*cos(2*x)
--------- - 8*cos(2*x)*log\\/ -1 + x / - ---------- - ----------
        3                                  -1 + x             2 
(-1 + x)                                              (-1 + x)

$$- 8 \log{\left(\sqrt{x - 1} \right)} \cos{\left(2 x \right)} - \frac{6 \sin{\left(2 x \right)}}{x - 1} - \frac{3 \cos{\left(2 x \right)}}{\left(x - 1\right)^{2}} + \frac{\sin{\left(2 x \right)}}{\left(x - 1\right)^{3}}$$

Simplify

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Derivative of sen(2x)ln((x-1)^(1/2))

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Piecewise:

The solution