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cos(ln^2x+3)

Derivative of cos(ln^2x+3)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   /   2       \
cos\log (x) + 3/
$$\cos{\left(\log{\left(x \right)}^{2} + 3 \right)}$$
d /   /   2       \\
--\cos\log (x) + 3//
dx                  
$$\frac{d}{d x} \cos{\left(\log{\left(x \right)}^{2} + 3 \right)}$$
Detail solution
  1. Let .

  2. The derivative of cosine is negative sine:

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

    1. Differentiate term by term:

      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:

      4. The derivative of the constant is zero.

      The result is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

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