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

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

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
            /  _______\
sin(2*x)*log\\/ x - 1 /
log⁡(x−1)sin⁡(2x)\log{\left(\sqrt{x - 1} \right)} \sin{\left(2 x \right)} 
sin(2*x)*log(sqrt(x - 1))
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=sin⁡(2x)f{\left(x \right)} = \sin{\left(2 x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=2xu = 2 x.

    2. The derivative of sine is cosine:

      ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

    3. Then, apply the chain rule. Multiply by ddx2x\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: xx goes to 11

        So, the result is: 22

      The result of the chain rule is:

      2cos⁡(2x)2 \cos{\left(2 x \right)}

    g(x)=log⁡(x−1)g{\left(x \right)} = \log{\left(\sqrt{x - 1} \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=x−1u = \sqrt{x - 1}.

    2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

    3. Then, apply the chain rule. Multiply by ddxx−1\frac{d}{d x} \sqrt{x - 1}:

      1. Let u=x−1u = x - 1.

      2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

      3. Then, apply the chain rule. Multiply by ddx(x−1)\frac{d}{d x} \left(x - 1\right):

        1. Differentiate x−1x - 1 term by term:

          1. Apply the power rule: xx goes to 11

          2. The derivative of the constant −1-1 is zero.

          The result is: 11

        The result of the chain rule is:

        12x−1\frac{1}{2 \sqrt{x - 1}}

      The result of the chain rule is:

      12(x−1)\frac{1}{2 \left(x - 1\right)}

    The result is: 2log⁡(x−1)cos⁡(2x)+sin⁡(2x)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)}

  2. Now simplify:

    (x−1)log⁡(x−1)cos⁡(2x)+sin⁡(2x)2x−1\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:

(x−1)log⁡(x−1)cos⁡(2x)+sin⁡(2x)2x−1\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
02468-8-6-4-2-1010-510
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The first derivative [src] 
 sin(2*x)                 /  _______\
--------- + 2*cos(2*x)*log\\/ x - 1 /
2*(x - 1)
2log⁡(x−1)cos⁡(2x)+sin⁡(2x)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)
−4log⁡(x−1)sin⁡(2x)+2cos⁡(2x)x−1−sin⁡(2x)2(x−1)2- 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)
−8log⁡(x−1)cos⁡(2x)−6sin⁡(2x)x−1−3cos⁡(2x)(x−1)2+sin⁡(2x)(x−1)3- 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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