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Identical expressions
(sin(five *x)^ three)/(ln(2x- three))
( sinus of (5 multiply by x) cubed ) divide by (ln(2x minus 3))
( sinus of (five multiply by x) to the power of three) divide by (ln(2x minus three))
(sin(5*x)3)/(ln(2x-3))
sin5*x3/ln2x-3
(sin(5*x)³)/(ln(2x-3))
(sin(5*x) to the power of 3)/(ln(2x-3))
(sin(5x)^3)/(ln(2x-3))
(sin(5x)3)/(ln(2x-3))
sin5x3/ln2x-3
sin5x^3/ln2x-3
(sin(5*x)^3) divide by (ln(2x-3))
Similar expressions
(sin(5*x)^3)/(ln(2x+3))

The derivative of the function/ (sin(5*x)^3)/(ln(2x-3))

Derivative of (sin(5*x)^3)/(ln(2x-3))

The solution

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    3       
 sin (5*x)  
------------
log(2*x - 3)

$$\frac{\sin^{3}{\left(5 x \right)}}{\log{\left(2 x - 3 \right)}}$$

  /    3       \
d | sin (5*x)  |
--|------------|
dx\log(2*x - 3)/

$$\frac{d}{d x} \frac{\sin^{3}{\left(5 x \right)}}{\log{\left(2 x - 3 \right)}}$$

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin^{3}{\left(5 x \right)}$ and $g{\left(x \right)} = \log{\left(2 x - 3 \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(5 x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(5 x \right)}$ :
  1. Let $u = 5 x$ .
  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 $\frac{d}{d x} 5 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $5$
    The result of the chain rule is:
    
    $5 \cos{\left(5 x \right)}$
  The result of the chain rule is:
  
  $15 \sin^{2}{\left(5 x \right)} \cos{\left(5 x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x - 3$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x - 3\right)$ :
  1. Differentiate $2 x - 3$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2$
    2. The derivative of the constant $\left(-1\right) 3$ is zero.
    The result is: $2$
  The result of the chain rule is:
  
  $\frac{2}{2 x - 3}$
Now plug in to the quotient rule:

$\frac{15 \log{\left(2 x - 3 \right)} \sin^{2}{\left(5 x \right)} \cos{\left(5 x \right)} - \frac{2 \sin^{3}{\left(5 x \right)}}{2 x - 3}}{\log{\left(2 x - 3 \right)}^{2}}$
Now simplify:

$\frac{\left(15 \cdot \left(2 x - 3\right) \log{\left(2 x - 3 \right)} \cos{\left(5 x \right)} - 2 \sin{\left(5 x \right)}\right) \sin^{2}{\left(5 x \right)}}{\left(2 x - 3\right) \log{\left(2 x - 3 \right)}^{2}}$

The answer is:

$\frac{\left(15 \cdot \left(2 x - 3\right) \log{\left(2 x - 3 \right)} \cos{\left(5 x \right)} - 2 \sin{\left(5 x \right)}\right) \sin^{2}{\left(5 x \right)}}{\left(2 x - 3\right) \log{\left(2 x - 3 \right)}^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

             3                    2              
        2*sin (5*x)         15*sin (5*x)*cos(5*x)
- ----------------------- + ---------------------
               2                 log(2*x - 3)    
  (2*x - 3)*log (2*x - 3)

$$\frac{15 \sin^{2}{\left(5 x \right)} \cos{\left(5 x \right)}}{\log{\left(2 x - 3 \right)}} - \frac{2 \sin^{3}{\left(5 x \right)}}{\left(2 x - 3\right) \log{\left(2 x - 3 \right)}^{2}}$$

Simplify

The second derivative [src]

/                                                                 2      /          2      \\         
|                                                            4*sin (5*x)*|1 + -------------||         
|        2               2          60*cos(5*x)*sin(5*x)                 \    log(-3 + 2*x)/|         
|- 75*sin (5*x) + 150*cos (5*x) - ------------------------ + -------------------------------|*sin(5*x)
|                                 (-3 + 2*x)*log(-3 + 2*x)                2                 |         
\                                                               (-3 + 2*x) *log(-3 + 2*x)   /         
------------------------------------------------------------------------------------------------------
                                            log(-3 + 2*x)

$$\frac{\left(\frac{4 \cdot \left(1 + \frac{2}{\log{\left(2 x - 3 \right)}}\right) \sin^{2}{\left(5 x \right)}}{\left(2 x - 3\right)^{2} \log{\left(2 x - 3 \right)}} - 75 \sin^{2}{\left(5 x \right)} + 150 \cos^{2}{\left(5 x \right)} - \frac{60 \sin{\left(5 x \right)} \cos{\left(5 x \right)}}{\left(2 x - 3\right) \log{\left(2 x - 3 \right)}}\right) \sin{\left(5 x \right)}}{\log{\left(2 x - 3 \right)}}$$

Simplify

The third derivative [src]

                                                     3      /          3               3       \                                                                                      
                                               16*sin (5*x)*|1 + ------------- + --------------|                                                   2      /          2      \         
                                                            |    log(-3 + 2*x)      2          |       /   2             2     \            180*sin (5*x)*|1 + -------------|*cos(5*x)
      /       2             2     \                         \                    log (-3 + 2*x)/   450*\sin (5*x) - 2*cos (5*x)/*sin(5*x)                 \    log(-3 + 2*x)/         
- 375*\- 2*cos (5*x) + 7*sin (5*x)/*cos(5*x) - ------------------------------------------------- + -------------------------------------- + ------------------------------------------
                                                                     3                                    (-3 + 2*x)*log(-3 + 2*x)                            2                       
                                                           (-3 + 2*x) *log(-3 + 2*x)                                                                (-3 + 2*x) *log(-3 + 2*x)         
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                    log(-3 + 2*x)

$$\frac{\frac{180 \cdot \left(1 + \frac{2}{\log{\left(2 x - 3 \right)}}\right) \sin^{2}{\left(5 x \right)} \cos{\left(5 x \right)}}{\left(2 x - 3\right)^{2} \log{\left(2 x - 3 \right)}} - 375 \cdot \left(7 \sin^{2}{\left(5 x \right)} - 2 \cos^{2}{\left(5 x \right)}\right) \cos{\left(5 x \right)} + \frac{450 \left(\sin^{2}{\left(5 x \right)} - 2 \cos^{2}{\left(5 x \right)}\right) \sin{\left(5 x \right)}}{\left(2 x - 3\right) \log{\left(2 x - 3 \right)}} - \frac{16 \cdot \left(1 + \frac{3}{\log{\left(2 x - 3 \right)}} + \frac{3}{\log{\left(2 x - 3 \right)}^{2}}\right) \sin^{3}{\left(5 x \right)}}{\left(2 x - 3\right)^{3} \log{\left(2 x - 3 \right)}}}{\log{\left(2 x - 3 \right)}}$$

Simplify

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Derivative of (sin(5*x)^3)/(ln(2x-3))

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The solution