Derivative of (sinx)/(ln(x+2))

The solution

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  sin(x)  
----------
log(x + 2)

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

sin(x)/log(x + 2)

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                               /        2     \       
                               |1 + ----------|*sin(x)
               2*cos(x)        \    log(2 + x)/       
-sin(x) - ------------------ + -----------------------
          (2 + x)*log(2 + x)            2             
                                 (2 + x) *log(2 + x)  
------------------------------------------------------
                      log(2 + x)

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (sinx)/(ln(x+2))

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The solution