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Derivative of (ln^2)(sin(7^x+2))

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   2       / x    \
log (x)*sin\7  + 2/
$$\log{\left(x \right)}^{2} \sin{\left(7^{x} + 2 \right)}$$
log(x)^2*sin(7^x + 2)
Detail solution
  1. Apply the product rule:

    ; to find :

    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:

    ; to find :

    1. Let .

    2. The derivative of sine is cosine:

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

      1. Differentiate term by term:

        1. The derivative of the constant is zero.

        The result is:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
            / x    \                                
2*log(x)*sin\7  + 2/    x    2       / x    \       
-------------------- + 7 *log (x)*cos\7  + 2/*log(7)
         x                                          
$$7^{x} \log{\left(7 \right)} \log{\left(x \right)}^{2} \cos{\left(7^{x} + 2 \right)} + \frac{2 \log{\left(x \right)} \sin{\left(7^{x} + 2 \right)}}{x}$$
The second derivative [src]
                     /     x\                                                            x    /     x\              
  2*(-1 + log(x))*sin\2 + 7 /    x    2       2    /     /     x\    x    /     x\\   4*7 *cos\2 + 7 /*log(7)*log(x)
- --------------------------- - 7 *log (7)*log (x)*\- cos\2 + 7 / + 7 *sin\2 + 7 // + ------------------------------
                2                                                                                   x               
               x                                                                                                    
$$- 7^{x} \left(7^{x} \sin{\left(7^{x} + 2 \right)} - \cos{\left(7^{x} + 2 \right)}\right) \log{\left(7 \right)}^{2} \log{\left(x \right)}^{2} + \frac{4 \cdot 7^{x} \log{\left(7 \right)} \log{\left(x \right)} \cos{\left(7^{x} + 2 \right)}}{x} - \frac{2 \left(\log{\left(x \right)} - 1\right) \sin{\left(7^{x} + 2 \right)}}{x^{2}}$$
The third derivative [src]
                     /     x\                                                                                 x    2    /     /     x\    x    /     x\\             x                  /     x\       
2*(-3 + 2*log(x))*sin\2 + 7 /    x    3       2    /     /     x\    2*x    /     x\      x    /     x\\   6*7 *log (7)*\- cos\2 + 7 / + 7 *sin\2 + 7 //*log(x)   6*7 *(-1 + log(x))*cos\2 + 7 /*log(7)
----------------------------- - 7 *log (7)*log (x)*\- cos\2 + 7 / + 7   *cos\2 + 7 / + 3*7 *sin\2 + 7 // - ---------------------------------------------------- - -------------------------------------
               3                                                                                                                    x                                                2                 
              x                                                                                                                                                                     x                  
$$- 7^{x} \left(7^{2 x} \cos{\left(7^{x} + 2 \right)} + 3 \cdot 7^{x} \sin{\left(7^{x} + 2 \right)} - \cos{\left(7^{x} + 2 \right)}\right) \log{\left(7 \right)}^{3} \log{\left(x \right)}^{2} - \frac{6 \cdot 7^{x} \left(7^{x} \sin{\left(7^{x} + 2 \right)} - \cos{\left(7^{x} + 2 \right)}\right) \log{\left(7 \right)}^{2} \log{\left(x \right)}}{x} - \frac{6 \cdot 7^{x} \left(\log{\left(x \right)} - 1\right) \log{\left(7 \right)} \cos{\left(7^{x} + 2 \right)}}{x^{2}} + \frac{2 \left(2 \log{\left(x \right)} - 3\right) \sin{\left(7^{x} + 2 \right)}}{x^{3}}$$