Derivative of sin(x/3)+lnx²

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

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

sin(x/3) + log(x)^2

Detail solution

Differentiate $\log{\left(x \right)}^{2} + \sin{\left(\frac{x}{3} \right)}$ term by term:
1. Let $u = \frac{x}{3}$ .
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 $\frac{d}{d x} \frac{x}{3}$ :
  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: $\frac{1}{3}$
  The result of the chain rule is:
  
  $\frac{\cos{\left(\frac{x}{3} \right)}}{3}$
4. Let $u = \log{\left(x \right)}$ .
5. Apply the power rule: $u^{2}$ goes to $2 u$
6. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result of the chain rule is:
  
  $\frac{2 \log{\left(x \right)}}{x}$
The result is: $\frac{\cos{\left(\frac{x}{3} \right)}}{3} + \frac{2 \log{\left(x \right)}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

        /x\           
     sin|-|           
2       \3/   2*log(x)
-- - ------ - --------
 2     9          2   
x                x

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

Simplify

The third derivative [src]

          /x\           
       cos|-|           
  6       \3/   4*log(x)
- -- - ------ + --------
   3     27         3   
  x                x

$$- \frac{\cos{\left(\frac{x}{3} \right)}}{27} + \frac{4 \log{\left(x \right)}}{x^{3}} - \frac{6}{x^{3}}$$

Simplify