Derivative of y=lg(x-cosx)

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log(x - cos(x))

$$\log{\left(x - \cos{\left(x \right)} \right)}$$

log(x - cos(x))

Detail solution

Let $u = x - \cos{\left(x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - \cos{\left(x \right)}\right)$ :
1. Differentiate $x - \cos{\left(x \right)}$ term by term:
  1. Apply the power rule: $x$ goes to $1$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    So, the result is: $\sin{\left(x \right)}$
  The result is: $\sin{\left(x \right)} + 1$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

1 + sin(x)
----------
x - cos(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=lg(x-cosx)

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