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Identical expressions
y=(ln8x*cos10x)/(5x)
y equally (ln8x multiply by co sinus of e of 10x) divide by (5x)
y=(ln8xcos10x)/(5x)
y=ln8xcos10x/5x
y=(ln8x*cos10x) divide by (5x)
Similar expressions
y=(ln8x*cos10x)/5x

The derivative of the function/ y=(ln8x*cos10x)/(5x)

Derivative of y=(ln8x*cos10x)/(5x)

The solution

You have entered [src]

log(8*x)*cos(10*x)
------------------
       5*x

$$\frac{\log{\left(8 x \right)} \cos{\left(10 x \right)}}{5 x}$$

d /log(8*x)*cos(10*x)\
--|------------------|
dx\       5*x        /

$$\frac{d}{d x} \frac{\log{\left(8 x \right)} \cos{\left(10 x \right)}}{5 x}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(8 x \right)} \cos{\left(10 x \right)}$ and $g{\left(x \right)} = 5 x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \cos{\left(10 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 10 x$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 10 x$ :
    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: $10$
    The result of the chain rule is:
    
    $- 10 \sin{\left(10 x \right)}$
  $g{\left(x \right)} = \log{\left(8 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 8 x$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 8 x$ :
    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: $8$
    The result of the chain rule is:
    
    $\frac{1}{x}$
  The result is: $- 10 \log{\left(8 x \right)} \sin{\left(10 x \right)} + \frac{\cos{\left(10 x \right)}}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
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: $5$
Now plug in to the quotient rule:

$\frac{5 x \left(- 10 \log{\left(8 x \right)} \sin{\left(10 x \right)} + \frac{\cos{\left(10 x \right)}}{x}\right) - 5 \log{\left(8 x \right)} \cos{\left(10 x \right)}}{25 x^{2}}$
Now simplify:

$\frac{- 10 x \log{\left(8 x \right)} \sin{\left(10 x \right)} - \log{\left(8 x \right)} \cos{\left(10 x \right)} + \cos{\left(10 x \right)}}{5 x^{2}}$

The answer is:

$\frac{- 10 x \log{\left(8 x \right)} \sin{\left(10 x \right)} - \log{\left(8 x \right)} \cos{\left(10 x \right)} + \cos{\left(10 x \right)}}{5 x^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 1                                                            
---*cos(10*x)                                                 
5*x                 1                       cos(10*x)*log(8*x)
------------- - 10*---*log(8*x)*sin(10*x) - ------------------
      x            5*x                                2       
                                                   5*x

$$- 10 \cdot \frac{1}{5 x} \log{\left(8 x \right)} \sin{\left(10 x \right)} + \frac{\frac{1}{5 x} \cos{\left(10 x \right)}}{x} - \frac{\log{\left(8 x \right)} \cos{\left(10 x \right)}}{5 x^{2}}$$

Simplify

The second derivative [src]

                         4*sin(10*x)   3*cos(10*x)   4*log(8*x)*sin(10*x)   2*cos(10*x)*log(8*x)
-20*cos(10*x)*log(8*x) - ----------- - ----------- + -------------------- + --------------------
                              x               2               x                        2        
                                           5*x                                      5*x         
------------------------------------------------------------------------------------------------
                                               x

$$\frac{- 20 \log{\left(8 x \right)} \cos{\left(10 x \right)} + \frac{4 \log{\left(8 x \right)} \sin{\left(10 x \right)}}{x} - \frac{4 \sin{\left(10 x \right)}}{x} + \frac{2 \log{\left(8 x \right)} \cos{\left(10 x \right)}}{5 x^{2}} - \frac{3 \cos{\left(10 x \right)}}{5 x^{2}}}{x}$$

Simplify

The third derivative [src]

  60*cos(10*x)   18*sin(10*x)                            11*cos(10*x)   12*log(8*x)*sin(10*x)   60*cos(10*x)*log(8*x)   6*cos(10*x)*log(8*x)
- ------------ + ------------ + 200*log(8*x)*sin(10*x) + ------------ - --------------------- + --------------------- - --------------------
       x               2                                        3                  2                      x                        3        
                      x                                      5*x                  x                                             5*x         
--------------------------------------------------------------------------------------------------------------------------------------------
                                                                     x

$$\frac{200 \log{\left(8 x \right)} \sin{\left(10 x \right)} + \frac{60 \log{\left(8 x \right)} \cos{\left(10 x \right)}}{x} - \frac{60 \cos{\left(10 x \right)}}{x} - \frac{12 \log{\left(8 x \right)} \sin{\left(10 x \right)}}{x^{2}} + \frac{18 \sin{\left(10 x \right)}}{x^{2}} - \frac{6 \log{\left(8 x \right)} \cos{\left(10 x \right)}}{5 x^{3}} + \frac{11 \cos{\left(10 x \right)}}{5 x^{3}}}{x}$$

Simplify

The graph

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Identical expressions

Similar expressions

Derivative of y=(ln8x*cos10x)/(5x)

The graph:

Piecewise:

The solution