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Limit of the function:
sin(3*x)/log(1+2*x)
Identical expressions
sin(three *x)/log(one + two *x)
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sinus of (three multiply by x) divide by logarithm of (one plus two multiply by x)
sin(3x)/log(1+2x)
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sin(3*x) divide by log(1+2*x)
Similar expressions
sin(3*x)/log(1-2*x)

The derivative of the function/ sin(3*x)/log(1+2*x)

Derivative of sin(3x)/log(1+2x)

The solution

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  sin(3*x)  
------------
log(1 + 2*x)

$$\frac{\sin{\left(3 x \right)}}{\log{\left(2 x + 1 \right)}}$$

sin(3*x)/log(1 + 2*x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(3 x \right)}$ and $g{\left(x \right)} = \log{\left(2 x + 1 \right)}$ .

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

$\frac{3 \log{\left(2 x + 1 \right)} \cos{\left(3 x \right)} - \frac{2 \sin{\left(3 x \right)}}{2 x + 1}}{\log{\left(2 x + 1 \right)}^{2}}$
Now simplify:

$\frac{3 \left(2 x + 1\right) \log{\left(2 x + 1 \right)} \cos{\left(3 x \right)} - 2 \sin{\left(3 x \right)}}{\left(2 x + 1\right) \log{\left(2 x + 1 \right)}^{2}}$

The answer is:

$\frac{3 \left(2 x + 1\right) \log{\left(2 x + 1 \right)} \cos{\left(3 x \right)} - 2 \sin{\left(3 x \right)}}{\left(2 x + 1\right) \log{\left(2 x + 1 \right)}^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 3*cos(3*x)           2*sin(3*x)      
------------ - -----------------------
log(1 + 2*x)                2         
               (1 + 2*x)*log (1 + 2*x)

$$\frac{3 \cos{\left(3 x \right)}}{\log{\left(2 x + 1 \right)}} - \frac{2 \sin{\left(3 x \right)}}{\left(2 x + 1\right) \log{\left(2 x + 1 \right)}^{2}}$$

Simplify

The second derivative [src]

                                         /         2      \         
                                       4*|1 + ------------|*sin(3*x)
                   12*cos(3*x)           \    log(1 + 2*x)/         
-9*sin(3*x) - ---------------------- + -----------------------------
              (1 + 2*x)*log(1 + 2*x)               2                
                                          (1 + 2*x) *log(1 + 2*x)   
--------------------------------------------------------------------
                            log(1 + 2*x)

$$\frac{\frac{4 \left(1 + \frac{2}{\log{\left(2 x + 1 \right)}}\right) \sin{\left(3 x \right)}}{\left(2 x + 1\right)^{2} \log{\left(2 x + 1 \right)}} - 9 \sin{\left(3 x \right)} - \frac{12 \cos{\left(3 x \right)}}{\left(2 x + 1\right) \log{\left(2 x + 1 \right)}}}{\log{\left(2 x + 1 \right)}}$$

Simplify

The third derivative [src]

                                           /         3               3      \                                          
                                        16*|1 + ------------ + -------------|*sin(3*x)      /         2      \         
                                           |    log(1 + 2*x)      2         |            36*|1 + ------------|*cos(3*x)
                    54*sin(3*x)            \                   log (1 + 2*x)/               \    log(1 + 2*x)/         
-27*cos(3*x) + ---------------------- - ---------------------------------------------- + ------------------------------
               (1 + 2*x)*log(1 + 2*x)                       3                                        2                 
                                                   (1 + 2*x) *log(1 + 2*x)                  (1 + 2*x) *log(1 + 2*x)    
-----------------------------------------------------------------------------------------------------------------------
                                                      log(1 + 2*x)

$$\frac{\frac{36 \left(1 + \frac{2}{\log{\left(2 x + 1 \right)}}\right) \cos{\left(3 x \right)}}{\left(2 x + 1\right)^{2} \log{\left(2 x + 1 \right)}} - 27 \cos{\left(3 x \right)} + \frac{54 \sin{\left(3 x \right)}}{\left(2 x + 1\right) \log{\left(2 x + 1 \right)}} - \frac{16 \left(1 + \frac{3}{\log{\left(2 x + 1 \right)}} + \frac{3}{\log{\left(2 x + 1 \right)}^{2}}\right) \sin{\left(3 x \right)}}{\left(2 x + 1\right)^{3} \log{\left(2 x + 1 \right)}}}{\log{\left(2 x + 1 \right)}}$$

Simplify

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Identical expressions

Similar expressions

Derivative of sin(3x)/log(1+2x)

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sin(3*x)/log(1+2*x)

The graph:

Piecewise:

The solution

Derivative of sin(3x)/log(1+2x)