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

You have entered [src]

x              
- + sin(log(x))
2

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

x/2 + sin(log(x))

Detail solution

Differentiate $\frac{x}{2} + \sin{\left(\log{\left(x \right)} \right)}$ 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: $\frac{1}{2}$
2. Let $u = \log{\left(x \right)}$ .
3. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
4. 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:
  
  $\frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$
The result is: $\frac{1}{2} + \frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

1   cos(log(x))
- + -----------
2        x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph