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sin(log(x)^(five))^(two)*x
sinus of ( logarithm of (x) to the power of (5)) to the power of (2) multiply by x
sinus of ( logarithm of (x) to the power of (five)) to the power of (two) multiply by x
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The derivative of the function/ sin(log(x)^(5))^(2)*x

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

The solution

You have entered [src]

   2/   5   \  
sin \log (x)/*x

$$x \sin^{2}{\left(\log{\left(x \right)}^{5} \right)}$$

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

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin^{2}{\left(\log{\left(x \right)}^{5} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(\log{\left(x \right)}^{5} \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(\log{\left(x \right)}^{5} \right)}$ :
  1. Let $u = \log{\left(x \right)}^{5}$ .
  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 $\frac{d}{d x} \log{\left(x \right)}^{5}$ :
    1. Let $u = \log{\left(x \right)}$ .
    2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
      1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      The result of the chain rule is:
      
      $\frac{5 \log{\left(x \right)}^{4}}{x}$
    The result of the chain rule is:
    
    $\frac{5 \log{\left(x \right)}^{4} \cos{\left(\log{\left(x \right)}^{5} \right)}}{x}$
  The result of the chain rule is:
  
  $\frac{10 \log{\left(x \right)}^{4} \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)}}{x}$
$g{\left(x \right)} = x$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
The result is: $10 \log{\left(x \right)}^{4} \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)} + \sin^{2}{\left(\log{\left(x \right)}^{5} \right)}$
Now simplify:

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

The answer is:

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

The first derivative [src]

   2/   5   \         4       /   5   \    /   5   \
sin \log (x)/ + 10*log (x)*cos\log (x)/*sin\log (x)/

$$10 \log{\left(x \right)}^{4} \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)} + \sin^{2}{\left(\log{\left(x \right)}^{5} \right)}$$

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

$$\frac{10 \left(- 5 \log{\left(x \right)}^{5} \sin^{2}{\left(\log{\left(x \right)}^{5} \right)} + 5 \log{\left(x \right)}^{5} \cos^{2}{\left(\log{\left(x \right)}^{5} \right)} + \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)} + 4 \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)}\right) \log{\left(x \right)}^{3}}{x}$$

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

$$- \frac{10 \left(3 \left(5 \log{\left(x \right)}^{5} \sin^{2}{\left(\log{\left(x \right)}^{5} \right)} - 5 \log{\left(x \right)}^{5} \cos^{2}{\left(\log{\left(x \right)}^{5} \right)} + \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)} - 4 \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)}\right) \log{\left(x \right)} + 100 \log{\left(x \right)}^{10} \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)} - 15 \log{\left(x \right)}^{6} \sin^{2}{\left(\log{\left(x \right)}^{5} \right)} + 15 \log{\left(x \right)}^{6} \cos^{2}{\left(\log{\left(x \right)}^{5} \right)} + 60 \log{\left(x \right)}^{5} \sin^{2}{\left(\log{\left(x \right)}^{5} \right)} - 60 \log{\left(x \right)}^{5} \cos^{2}{\left(\log{\left(x \right)}^{5} \right)} - 2 \log{\left(x \right)}^{2} \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)} + 12 \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)} - 12 \sin{\left(\log{\left(x \right)}^{5} \right)} \cos{\left(\log{\left(x \right)}^{5} \right)}\right) \log{\left(x \right)}^{2}}{x^{2}}$$

Simplify

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

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The solution