Derivative of ln(5x+cosx)

You have entered [src]

log(5*x + cos(x))

$$\log{\left(5 x + \cos{\left(x \right)} \right)}$$

log(5*x + cos(x))

Detail solution

Let $u = 5 x + \cos{\left(x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x + \cos{\left(x \right)}\right)$ :
1. Differentiate $5 x + \cos{\left(x \right)}$ 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: $5$
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result is: $5 - \sin{\left(x \right)}$
The result of the chain rule is:

$\frac{5 - \sin{\left(x \right)}}{5 x + \cos{\left(x \right)}}$

The answer is:

$\frac{5 - \sin{\left(x \right)}}{5 x + \cos{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 5 - sin(x) 
------------
5*x + cos(x)

$$\frac{5 - \sin{\left(x \right)}}{5 x + \cos{\left(x \right)}}$$

Simplify

The second derivative [src]

 /             2         \ 
 |(-5 + sin(x))          | 
-|-------------- + cos(x)| 
 \ 5*x + cos(x)          / 
---------------------------
        5*x + cos(x)

$$- \frac{\cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} - 5\right)^{2}}{5 x + \cos{\left(x \right)}}}{5 x + \cos{\left(x \right)}}$$

Simplify

The third derivative [src]

                 3                                  
  2*(-5 + sin(x))    3*(-5 + sin(x))*cos(x)         
- ---------------- - ---------------------- + sin(x)
                2         5*x + cos(x)              
  (5*x + cos(x))                                    
----------------------------------------------------
                    5*x + cos(x)

$$\frac{\sin{\left(x \right)} - \frac{3 \left(\sin{\left(x \right)} - 5\right) \cos{\left(x \right)}}{5 x + \cos{\left(x \right)}} - \frac{2 \left(\sin{\left(x \right)} - 5\right)^{3}}{\left(5 x + \cos{\left(x \right)}\right)^{2}}}{5 x + \cos{\left(x \right)}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of ln(5x+cosx)

The graph:

Piecewise:

The solution