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cos(ln^2x+ three)
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Similar expressions
cos(ln^2x-3)

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

Derivative of cos(ln^2x+3)

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

$$\frac{d}{d x} \cos{\left(\log{\left(x \right)}^{2} + 3 \right)}$$

Transform

Detail solution

Let $u = \log{\left(x \right)}^{2} + 3$ .
The derivative of cosine is negative sine:

$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\log{\left(x \right)}^{2} + 3\right)$ :
1. Differentiate $\log{\left(x \right)}^{2} + 3$ term by term:
  1. Let $u = \log{\left(x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  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:
    
    $\frac{2 \log{\left(x \right)}}{x}$
  4. The derivative of the constant $3$ is zero.
  The result is: $\frac{2 \log{\left(x \right)}}{x}$
The result of the chain rule is:

$- \frac{2 \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{2} + 3 \right)}}{x}$
Now simplify:

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

The answer is:

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

The first derivative [src]

             /   2       \
-2*log(x)*sin\log (x) + 3/
--------------------------
            x

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

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

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

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)//
----------------------------------------------------------------------------------------------------------------------------------------
                                                                    3                                                                   
                                                                   x

$$\frac{2 \cdot \left(4 \log{\left(x \right)}^{3} \sin{\left(\log{\left(x \right)}^{2} + 3 \right)} + 6 \log{\left(x \right)}^{2} \cos{\left(\log{\left(x \right)}^{2} + 3 \right)} - 2 \log{\left(x \right)} \sin{\left(\log{\left(x \right)}^{2} + 3 \right)} - 6 \log{\left(x \right)} \cos{\left(\log{\left(x \right)}^{2} + 3 \right)} + 3 \sin{\left(\log{\left(x \right)}^{2} + 3 \right)}\right)}{x^{3}}$$

Simplify

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Derivative of cos(ln^2x+3)

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The solution