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The derivative of the function/ sin(log(x)^(5))^(2)*x

Derivative of sin(log(x)^(5))^(2)*x

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
   2/   5   \  
sin \log (x)/*x
xsin⁡2(log⁡(x)5)x \sin^{2}{\left(\log{\left(x \right)}^{5} \right)} 
sin(log(x)^5)^2*x
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⁡2(log⁡(x)5)f{\left(x \right)} = \sin^{2}{\left(\log{\left(x \right)}^{5} \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=sin⁡(log⁡(x)5)u = \sin{\left(\log{\left(x \right)}^{5} \right)}.

    2. Apply the power rule: u2u^{2} goes to 2u2 u

    3. Then, apply the chain rule. Multiply by ddxsin⁡(log⁡(x)5)\frac{d}{d x} \sin{\left(\log{\left(x \right)}^{5} \right)}:

      1. Let u=log⁡(x)5u = \log{\left(x \right)}^{5}.

      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 ddxlog⁡(x)5\frac{d}{d x} \log{\left(x \right)}^{5}:

        1. Let u=log⁡(x)u = \log{\left(x \right)}.

        2. Apply the power rule: u5u^{5} goes to 5u45 u^{4}

        3. Then, apply the chain rule. Multiply by ddxlog⁡(x)\frac{d}{d x} \log{\left(x \right)}:

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

          The result of the chain rule is:

          5log⁡(x)4x\frac{5 \log{\left(x \right)}^{4}}{x}

        The result of the chain rule is:

        5log⁡(x)4cos⁡(log⁡(x)5)x\frac{5 \log{\left(x \right)}^{4} \cos{\left(\log{\left(x \right)}^{5} \right)}}{x}

      The result of the chain rule is:

      10log⁡(x)4sin⁡(log⁡(x)5)cos⁡(log⁡(x)5)x\frac{10 \log{\left(x \right)}^{4} \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)}}{x}

    g(x)=xg{\left(x \right)} = x; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Apply the power rule: xx goes to 11

    The result is: 10log⁡(x)4sin⁡(log⁡(x)5)cos⁡(log⁡(x)5)+sin⁡2(log⁡(x)5)10 \log{\left(x \right)}^{4} \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)} + \sin^{2}{\left(\log{\left(x \right)}^{5} \right)}

  2. Now simplify:

    5log⁡(x)4sin⁡(2log⁡(x)5)−cos⁡(2log⁡(x)5)2+125 \log{\left(x \right)}^{4} \sin{\left(2 \log{\left(x \right)}^{5} \right)} - \frac{\cos{\left(2 \log{\left(x \right)}^{5} \right)}}{2} + \frac{1}{2}

The answer is:

5log⁡(x)4sin⁡(2log⁡(x)5)−cos⁡(2log⁡(x)5)2+125 \log{\left(x \right)}^{4} \sin{\left(2 \log{\left(x \right)}^{5} \right)} - \frac{\cos{\left(2 \log{\left(x \right)}^{5} \right)}}{2} + \frac{1}{2}

The first derivative [src] 
   2/   5   \         4       /   5   \    /   5   \
sin \log (x)/ + 10*log (x)*cos\log (x)/*sin\log (x)/
10log⁡(x)4sin⁡(log⁡(x)5)cos⁡(log⁡(x)5)+sin⁡2(log⁡(x)5)10 \log{\left(x \right)}^{4} \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)} + \sin^{2}{\left(\log{\left(x \right)}^{5} \right)}
Simplify
The second derivative [src] 
      3    /       5       2/   5   \        /   5   \    /   5   \        2/   5   \    5         /   5   \           /   5   \\
10*log (x)*\- 5*log (x)*sin \log (x)/ + 4*cos\log (x)/*sin\log (x)/ + 5*cos \log (x)/*log (x) + cos\log (x)/*log(x)*sin\log (x)//
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                                                                x
10(−5log⁡(x)5sin⁡2(log⁡(x)5)+5log⁡(x)5cos⁡2(log⁡(x)5)+log⁡(x)sin⁡(log⁡(x)5)cos⁡(log⁡(x)5)+4sin⁡(log⁡(x)5)cos⁡(log⁡(x)5))log⁡(x)3x\frac{10 \left(- 5 \log{\left(x \right)}^{5} \sin^{2}{\left(\log{\left(x \right)}^{5} \right)} + 5 \log{\left(x \right)}^{5} \cos^{2}{\left(\log{\left(x \right)}^{5} \right)} + \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)} + 4 \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)}\right) \log{\left(x \right)}^{3}}{x}
Simplify
The third derivative [src] 
       2    /        2/   5   \    5            6       2/   5   \         /   5   \    /   5   \     /       2/   5   \    5           /   5   \    /   5   \        5       2/   5   \      /   5   \           /   5   \\                2/   5   \    6            5       2/   5   \        2       /   5   \    /   5   \         /   5   \           /   5   \          10       /   5   \    /   5   \\
-10*log (x)*\- 60*cos \log (x)/*log (x) - 15*log (x)*sin \log (x)/ - 12*cos\log (x)/*sin\log (x)/ + 3*\- 5*cos \log (x)/*log (x) - 4*cos\log (x)/*sin\log (x)/ + 5*log (x)*sin \log (x)/ + cos\log (x)/*log(x)*sin\log (x)//*log(x) + 15*cos \log (x)/*log (x) + 60*log (x)*sin \log (x)/ - 2*log (x)*cos\log (x)/*sin\log (x)/ + 12*cos\log (x)/*log(x)*sin\log (x)/ + 100*log  (x)*cos\log (x)/*sin\log (x)//
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                                                                                                                                                                                                        2                                                                                                                                                                                                      
                                                                                                                                                                                                       x
−10(3(5log⁡(x)5sin⁡2(log⁡(x)5)−5log⁡(x)5cos⁡2(log⁡(x)5)+log⁡(x)sin⁡(log⁡(x)5)cos⁡(log⁡(x)5)−4sin⁡(log⁡(x)5)cos⁡(log⁡(x)5))log⁡(x)+100log⁡(x)10sin⁡(log⁡(x)5)cos⁡(log⁡(x)5)−15log⁡(x)6sin⁡2(log⁡(x)5)+15log⁡(x)6cos⁡2(log⁡(x)5)+60log⁡(x)5sin⁡2(log⁡(x)5)−60log⁡(x)5cos⁡2(log⁡(x)5)−2log⁡(x)2sin⁡(log⁡(x)5)cos⁡(log⁡(x)5)+12log⁡(x)sin⁡(log⁡(x)5)cos⁡(log⁡(x)5)−12sin⁡(log⁡(x)5)cos⁡(log⁡(x)5))log⁡(x)2x2- \frac{10 \left(3 \left(5 \log{\left(x \right)}^{5} \sin^{2}{\left(\log{\left(x \right)}^{5} \right)} - 5 \log{\left(x \right)}^{5} \cos^{2}{\left(\log{\left(x \right)}^{5} \right)} + \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)} - 4 \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)}\right) \log{\left(x \right)} + 100 \log{\left(x \right)}^{10} \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)} - 15 \log{\left(x \right)}^{6} \sin^{2}{\left(\log{\left(x \right)}^{5} \right)} + 15 \log{\left(x \right)}^{6} \cos^{2}{\left(\log{\left(x \right)}^{5} \right)} + 60 \log{\left(x \right)}^{5} \sin^{2}{\left(\log{\left(x \right)}^{5} \right)} - 60 \log{\left(x \right)}^{5} \cos^{2}{\left(\log{\left(x \right)}^{5} \right)} - 2 \log{\left(x \right)}^{2} \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)} + 12 \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)} - 12 \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)}\right) \log{\left(x \right)}^{2}}{x^{2}}
Simplify
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