Mister Exam

Derivative of ln(ax+b)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
log(a*x + b)
log(ax+b)\log{\left(a x + b \right)}
log(a*x + b)
Detail solution
  1. Let u=ax+bu = a x + b.

  2. The derivative of log(u)\log{\left(u \right)} is 1u\frac{1}{u}.

  3. Then, apply the chain rule. Multiply by x(ax+b)\frac{\partial}{\partial x} \left(a x + b\right):

    1. Differentiate ax+ba 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: xx goes to 11

        So, the result is: aa

      2. The derivative of the constant bb is zero.

      The result is: aa

    The result of the chain rule is:

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

  4. Now simplify:

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


The answer is:

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

The first derivative [src]
   a   
-------
a*x + b
aax+b\frac{a}{a x + b}
The second derivative [src]
     2    
   -a     
----------
         2
(b + a*x) 
a2(ax+b)2- \frac{a^{2}}{\left(a x + b\right)^{2}}
The third derivative [src]
      3   
   2*a    
----------
         3
(b + a*x) 
2a3(ax+b)3\frac{2 a^{3}}{\left(a x + b\right)^{3}}