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The derivative of the function/ (ax+b)^n

Derivative of (ax+b)^n

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
         n
(a*x + b)
$$\left(a x + b\right)^{n}$$ 
(a*x + b)^n
Detail solution

  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. Differentiate 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: goes to

        So, the result is:

      2. The derivative of the constant is zero.

      The result is:

    The result of the chain rule is:

  4. Now simplify:

The answer is:

The first derivative [src] 
             n
a*n*(a*x + b) 
--------------
   a*x + b
$$\frac{a n \left(a x + b\right)^{n}}{a x + b}$$
Simplify
The second derivative [src] 
   2          n         
n*a *(b + a*x) *(-1 + n)
------------------------
                2       
       (b + a*x)
$$\frac{a^{2} n \left(n - 1\right) \left(a x + b\right)^{n}}{\left(a x + b\right)^{2}}$$
Simplify
The third derivative [src] 
   3          n /     2      \
n*a *(b + a*x) *\2 + n  - 3*n/
------------------------------
                   3          
          (b + a*x)
$$\frac{a^{3} n \left(a x + b\right)^{n} \left(n^{2} - 3 n + 2\right)}{\left(a x + b\right)^{3}}$$
Simplify
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