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cos(ln^2x+3)
The derivative of the function/ cos(ln^2x+3)

Derivative of cos(ln^2x+3)

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

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The solution

You have entered [src] 
   /   2       \
cos\log (x) + 3/
cos⁡(log⁡(x)2+3)\cos{\left(\log{\left(x \right)}^{2} + 3 \right)} 
d /   /   2       \\
--\cos\log (x) + 3//
dx
ddxcos⁡(log⁡(x)2+3)\frac{d}{d x} \cos{\left(\log{\left(x \right)}^{2} + 3 \right)}
Transform
Detail solution

  1. Let u=log⁡(x)2+3u = \log{\left(x \right)}^{2} + 3.

  2. The derivative of cosine is negative sine:

    dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

  3. Then, apply the chain rule. Multiply by ddx(log⁡(x)2+3)\frac{d}{d x} \left(\log{\left(x \right)}^{2} + 3\right):

    1. Differentiate log⁡(x)2+3\log{\left(x \right)}^{2} + 3 term by term:

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

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

      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:

        2log⁡(x)x\frac{2 \log{\left(x \right)}}{x}

      4. The derivative of the constant 33 is zero.

      The result is: 2log⁡(x)x\frac{2 \log{\left(x \right)}}{x}

    The result of the chain rule is:

    −2log⁡(x)sin⁡(log⁡(x)2+3)x- \frac{2 \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{2} + 3 \right)}}{x}

  4. Now simplify:

    −2log⁡(x)sin⁡(log⁡(x)2+3)x- \frac{2 \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{2} + 3 \right)}}{x}

The answer is:

−2log⁡(x)sin⁡(log⁡(x)2+3)x- \frac{2 \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{2} + 3 \right)}}{x}

The first derivative [src] 
             /   2       \
-2*log(x)*sin\log (x) + 3/
--------------------------
            x
−2log⁡(x)sin⁡(log⁡(x)2+3)x- \frac{2 \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{2} + 3 \right)}}{x}
Simplify
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
2(−2log⁡(x)2cos⁡(log⁡(x)2+3)+log⁡(x)sin⁡(log⁡(x)2+3)−sin⁡(log⁡(x)2+3))x2\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}}
Simplify
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)//
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                                                                    3                                                                   
                                                                   x
2⋅(4log⁡(x)3sin⁡(log⁡(x)2+3)+6log⁡(x)2cos⁡(log⁡(x)2+3)−2log⁡(x)sin⁡(log⁡(x)2+3)−6log⁡(x)cos⁡(log⁡(x)2+3)+3sin⁡(log⁡(x)2+3))x3\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}}
Simplify
The graph
Derivative of cos(ln^2x+3)
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