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sin^ three (7x)/ln(4x+ five)
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Similar expressions
sin^3(7x)/ln(4x-5)

The derivative of the function/ sin^3(7x)/ln(4x+5)

Derivative of sin^3(7x)/ln(4x+5)

The solution

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    3       
 sin (7*x)  
------------
log(4*x + 5)

$$\frac{\sin^{3}{\left(7 x \right)}}{\log{\left(4 x + 5 \right)}}$$

sin(7*x)^3/log(4*x + 5)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin^{3}{\left(7 x \right)}$ and $g{\left(x \right)} = \log{\left(4 x + 5 \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(7 x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(7 x \right)}$ :
  1. Let $u = 7 x$ .
  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 $\frac{d}{d x} 7 x$ :
    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: $7$
    The result of the chain rule is:
    
    $7 \cos{\left(7 x \right)}$
  The result of the chain rule is:
  
  $21 \sin^{2}{\left(7 x \right)} \cos{\left(7 x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 4 x + 5$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 x + 5\right)$ :
  1. Differentiate $4 x + 5$ 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: $4$
    2. The derivative of the constant $5$ is zero.
    The result is: $4$
  The result of the chain rule is:
  
  $\frac{4}{4 x + 5}$
Now plug in to the quotient rule:

$\frac{21 \log{\left(4 x + 5 \right)} \sin^{2}{\left(7 x \right)} \cos{\left(7 x \right)} - \frac{4 \sin^{3}{\left(7 x \right)}}{4 x + 5}}{\log{\left(4 x + 5 \right)}^{2}}$
Now simplify:

$\frac{\left(21 \left(4 x + 5\right) \log{\left(4 x + 5 \right)} \cos{\left(7 x \right)} - 4 \sin{\left(7 x \right)}\right) \sin^{2}{\left(7 x \right)}}{\left(4 x + 5\right) \log{\left(4 x + 5 \right)}^{2}}$

The answer is:

$\frac{\left(21 \left(4 x + 5\right) \log{\left(4 x + 5 \right)} \cos{\left(7 x \right)} - 4 \sin{\left(7 x \right)}\right) \sin^{2}{\left(7 x \right)}}{\left(4 x + 5\right) \log{\left(4 x + 5 \right)}^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

             3                    2              
        4*sin (7*x)         21*sin (7*x)*cos(7*x)
- ----------------------- + ---------------------
               2                 log(4*x + 5)    
  (4*x + 5)*log (4*x + 5)

$$\frac{21 \sin^{2}{\left(7 x \right)} \cos{\left(7 x \right)}}{\log{\left(4 x + 5 \right)}} - \frac{4 \sin^{3}{\left(7 x \right)}}{\left(4 x + 5\right) \log{\left(4 x + 5 \right)}^{2}}$$

Simplify

The second derivative [src]

/                                                                 2      /         2      \\         
|                                                           16*sin (7*x)*|1 + ------------||         
|         2               2        168*cos(7*x)*sin(7*x)                 \    log(5 + 4*x)/|         
|- 147*sin (7*x) + 294*cos (7*x) - ---------------------- + -------------------------------|*sin(7*x)
|                                  (5 + 4*x)*log(5 + 4*x)                2                 |         
\                                                               (5 + 4*x) *log(5 + 4*x)    /         
-----------------------------------------------------------------------------------------------------
                                             log(5 + 4*x)

$$\frac{\left(\frac{16 \left(1 + \frac{2}{\log{\left(4 x + 5 \right)}}\right) \sin^{2}{\left(7 x \right)}}{\left(4 x + 5\right)^{2} \log{\left(4 x + 5 \right)}} - 147 \sin^{2}{\left(7 x \right)} + 294 \cos^{2}{\left(7 x \right)} - \frac{168 \sin{\left(7 x \right)} \cos{\left(7 x \right)}}{\left(4 x + 5\right) \log{\left(4 x + 5 \right)}}\right) \sin{\left(7 x \right)}}{\log{\left(4 x + 5 \right)}}$$

Simplify

The third derivative [src]

                                                       3      /         3               3      \                                                                                       
                                                128*sin (7*x)*|1 + ------------ + -------------|                                                     2      /         2      \         
                                                              |    log(5 + 4*x)      2         |        /   2             2     \            1008*sin (7*x)*|1 + ------------|*cos(7*x)
       /       2             2     \                          \                   log (5 + 4*x)/   1764*\sin (7*x) - 2*cos (7*x)/*sin(7*x)                  \    log(5 + 4*x)/         
- 1029*\- 2*cos (7*x) + 7*sin (7*x)/*cos(7*x) - ------------------------------------------------ + --------------------------------------- + ------------------------------------------
                                                                     3                                      (5 + 4*x)*log(5 + 4*x)                             2                       
                                                            (5 + 4*x) *log(5 + 4*x)                                                                   (5 + 4*x) *log(5 + 4*x)          
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                      log(5 + 4*x)

$$\frac{\frac{1008 \left(1 + \frac{2}{\log{\left(4 x + 5 \right)}}\right) \sin^{2}{\left(7 x \right)} \cos{\left(7 x \right)}}{\left(4 x + 5\right)^{2} \log{\left(4 x + 5 \right)}} - 1029 \left(7 \sin^{2}{\left(7 x \right)} - 2 \cos^{2}{\left(7 x \right)}\right) \cos{\left(7 x \right)} + \frac{1764 \left(\sin^{2}{\left(7 x \right)} - 2 \cos^{2}{\left(7 x \right)}\right) \sin{\left(7 x \right)}}{\left(4 x + 5\right) \log{\left(4 x + 5 \right)}} - \frac{128 \left(1 + \frac{3}{\log{\left(4 x + 5 \right)}} + \frac{3}{\log{\left(4 x + 5 \right)}^{2}}\right) \sin^{3}{\left(7 x \right)}}{\left(4 x + 5\right)^{3} \log{\left(4 x + 5 \right)}}}{\log{\left(4 x + 5 \right)}}$$

Simplify

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Derivative of sin^3(7x)/ln(4x+5)

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