Derivative of y=lnsin³5x+ln²x

The solution

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   35              2   
log  (sin(x)) + log (x)

$$\log{\left(x \right)}^{2} + \log{\left(\sin{\left(x \right)} \right)}^{35}$$

log(sin(x))^35 + log(x)^2

Detail solution

Differentiate $\log{\left(x \right)}^{2} + \log{\left(\sin{\left(x \right)} \right)}^{35}$ term by term:
1. Let $u = \log{\left(\sin{\left(x \right)} \right)}$ .
2. Apply the power rule: $u^{35}$ goes to $35 u^{34}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(\sin{\left(x \right)} \right)}$ :
  1. Let $u = \sin{\left(x \right)}$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    The result of the chain rule is:
    
    $\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
  The result of the chain rule is:
  
  $\frac{35 \log{\left(\sin{\left(x \right)} \right)}^{34} \cos{\left(x \right)}}{\sin{\left(x \right)}}$
4. Let $u = \log{\left(x \right)}$ .
5. Apply the power rule: $u^{2}$ goes to $2 u$
6. 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:
  
  $\frac{2 \log{\left(x \right)}}{x}$
The result is: $\frac{35 \log{\left(\sin{\left(x \right)} \right)}^{34} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 \log{\left(x \right)}}{x}$
Now simplify:

$\frac{35 \log{\left(\sin{\left(x \right)} \right)}^{34}}{\tan{\left(x \right)}} + \frac{2 \log{\left(x \right)}}{x}$

The answer is:

$\frac{35 \log{\left(\sin{\left(x \right)} \right)}^{34}}{\tan{\left(x \right)}} + \frac{2 \log{\left(x \right)}}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                 34               
2*log(x)   35*log  (sin(x))*cos(x)
-------- + -----------------------
   x                sin(x)

$$\frac{35 \log{\left(\sin{\left(x \right)} \right)}^{34} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 \log{\left(x \right)}}{x}$$

Simplify

The second derivative [src]

                                           2       34                   2       33        
        34           2    2*log(x)   35*cos (x)*log  (sin(x))   1190*cos (x)*log  (sin(x))
- 35*log  (sin(x)) + -- - -------- - ------------------------ + --------------------------
                      2       2                 2                           2             
                     x       x               sin (x)                     sin (x)

$$- 35 \log{\left(\sin{\left(x \right)} \right)}^{34} - \frac{35 \log{\left(\sin{\left(x \right)} \right)}^{34} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{1190 \log{\left(\sin{\left(x \right)} \right)}^{33} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \log{\left(x \right)}}{x^{2}} + \frac{2}{x^{2}}$$

Simplify

The third derivative [src]

  /                          3       33                   33                        3       34                 34                           3       32        \
  |  3    2*log(x)   1785*cos (x)*log  (sin(x))   1785*log  (sin(x))*cos(x)   35*cos (x)*log  (sin(x))   35*log  (sin(x))*cos(x)   19635*cos (x)*log  (sin(x))|
2*|- -- + -------- - -------------------------- - ------------------------- + ------------------------ + ----------------------- + ---------------------------|
  |   3       3                  3                          sin(x)                       3                        sin(x)                        3             |
  \  x       x                sin (x)                                                 sin (x)                                                sin (x)          /

$$2 \left(\frac{35 \log{\left(\sin{\left(x \right)} \right)}^{34} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{35 \log{\left(\sin{\left(x \right)} \right)}^{34} \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} - \frac{1785 \log{\left(\sin{\left(x \right)} \right)}^{33} \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{1785 \log{\left(\sin{\left(x \right)} \right)}^{33} \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} + \frac{19635 \log{\left(\sin{\left(x \right)} \right)}^{32} \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} + \frac{2 \log{\left(x \right)}}{x^{3}} - \frac{3}{x^{3}}\right)$$

Simplify

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Derivative of y=lnsin³5x+ln²x

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