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The derivative of the function/ ln(ax+b)

Derivative of ln(ax+b)

The solution

You have entered [src]

log(a*x + b)

$$\log{\left(a x + b \right)}$$

log(a*x + b)

Detail solution

Let $u = a x + b$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} \left(a x + b\right)$ :
1. Differentiate $a x + b$ 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: $a$
  2. The derivative of the constant $b$ is zero.
  The result is: $a$
The result of the chain rule is:

$\frac{a}{a x + b}$
Now simplify:

$\frac{a}{a x + b}$

The answer is:

$\frac{a}{a x + b}$

The first derivative [src]

   a   
-------
a*x + b

$$\frac{a}{a x + b}$$

Simplify

The second derivative [src]

     2    
   -a     
----------
         2
(b + a*x)

$$- \frac{a^{2}}{\left(a x + b\right)^{2}}$$

Simplify

The third derivative [src]

      3   
   2*a    
----------
         3
(b + a*x)

$$\frac{2 a^{3}}{\left(a x + b\right)^{3}}$$

Simplify