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Identical expressions
(ln^ two)(sin(seven ^x+ two))
(ln squared )( sinus of (7 to the power of x plus 2))
(ln to the power of two)( sinus of (seven to the power of x plus two))
(ln2)(sin(7x+2))
ln2sin7x+2
(ln²)(sin(7^x+2))
(ln to the power of 2)(sin(7 to the power of x+2))
ln^2sin7^x+2
Similar expressions
(ln^2)(sin(7^x-2))

The derivative of the function/ (ln^2)(sin(7^x+2))

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

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

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \log{\left(x \right)}^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
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:
  
  $\frac{2 \log{\left(x \right)}}{x}$
$g{\left(x \right)} = \sin{\left(7^{x} + 2 \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 7^{x} + 2$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(7^{x} + 2\right)$ :
  1. Differentiate $7^{x} + 2$ term by term:
    1. $\frac{d}{d x} 7^{x} = 7^{x} \log{\left(7 \right)}$
    2. The derivative of the constant $2$ is zero.
    The result is: $7^{x} \log{\left(7 \right)}$
  The result of the chain rule is:
  
  $7^{x} \log{\left(7 \right)} \cos{\left(7^{x} + 2 \right)}$
The result is: $7^{x} \log{\left(7 \right)} \log{\left(x \right)}^{2} \cos{\left(7^{x} + 2 \right)} + \frac{2 \log{\left(x \right)} \sin{\left(7^{x} + 2 \right)}}{x}$
Now simplify:

$\frac{\left(7^{x} x \log{\left(7 \right)} \log{\left(x \right)} \cos{\left(7^{x} + 2 \right)} + 2 \sin{\left(7^{x} + 2 \right)}\right) \log{\left(x \right)}}{x}$

The answer is:

\frac{\left(7^{x} x \log{\left(7 \right)} \log{\left(x \right)} \cos{\left(7^{x} + 2 \right)} + 2 \sin{\left(7^{x} + 2 \right)}\right) \log{\left(x \right)}}{x}

The graph

Plot the graph f(x)

Plot the graph f'(x)

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}

Simplify

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}}

Simplify

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}}

Simplify

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

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The solution