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cos(lnx^2)

Derivative of cos(lnx^2)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   /   2   \
cos\log (x)/
$$\cos{\left(\log{\left(x \right)}^{2} \right)}$$
cos(log(x)^2)
Detail solution
  1. Let .

  2. The derivative of cosine is negative sine:

  3. Then, apply the chain rule. Multiply by :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. The derivative of is .

      The result of the chain rule is:

    The result of the chain rule is:


The answer is:

The graph
The first derivative [src]
             /   2   \
-2*log(x)*sin\log (x)/
----------------------
          x           
$$- \frac{2 \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{2} \right)}}{x}$$
The second derivative [src]
  /     /   2   \             /   2   \        2       /   2   \\
2*\- sin\log (x)/ + log(x)*sin\log (x)/ - 2*log (x)*cos\log (x)//
-----------------------------------------------------------------
                                 2                               
                                x                                
$$\frac{2 \left(- 2 \log{\left(x \right)}^{2} \cos{\left(\log{\left(x \right)}^{2} \right)} + \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{2} \right)} - \sin{\left(\log{\left(x \right)}^{2} \right)}\right)}{x^{2}}$$
The third derivative [src]
  /     /   2   \        /   2   \                      /   2   \        3       /   2   \        2       /   2   \\
2*\3*sin\log (x)/ - 6*cos\log (x)/*log(x) - 2*log(x)*sin\log (x)/ + 4*log (x)*sin\log (x)/ + 6*log (x)*cos\log (x)//
--------------------------------------------------------------------------------------------------------------------
                                                          3                                                         
                                                         x                                                          
$$\frac{2 \left(4 \log{\left(x \right)}^{3} \sin{\left(\log{\left(x \right)}^{2} \right)} + 6 \log{\left(x \right)}^{2} \cos{\left(\log{\left(x \right)}^{2} \right)} - 2 \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{2} \right)} - 6 \log{\left(x \right)} \cos{\left(\log{\left(x \right)}^{2} \right)} + 3 \sin{\left(\log{\left(x \right)}^{2} \right)}\right)}{x^{3}}$$
The graph
Derivative of cos(lnx^2)