To find the derivative of the function [tex]\( (a n + b)^5 \)[/tex] from first principles, also known as the limit definition of the derivative, we proceed as follows:
1. Definition of the derivative:
The derivative of a function [tex]\( f(n) \)[/tex] at any point [tex]\( n \)[/tex] is given by:
[tex]\[
f'(n) = \lim_{h \to 0} \frac{f(n + h) - f(n)}{h}
\][/tex]
2. Substitute the given function:
Here, our function [tex]\( f(n) = (a n + b)^5 \)[/tex].
We need to compute [tex]\( f(n + h) \)[/tex]:
[tex]\[
f(n + h) = (a (n + h) + b)^5 = (a n + a h + b)^5
\][/tex]
3. Formulate the difference quotient:
We then substitute [tex]\( f(n + h) \)[/tex] and [tex]\( f(n) \)[/tex] into the difference quotient:
[tex]\[
\frac{f(n + h) - f(n)}{h} = \frac{(a n + a h + b)^5 - (a n + b)^5}{h}
\][/tex]
4. Expand [tex]\( (a n + a h + b)^5 \)[/tex] using the Binomial Theorem:
The Binomial Theorem states:
[tex]\[
(x + y)^5 = \sum_{k=0}^{5} \binom{5}{k} x^{5-k} y^k
\][/tex]
In our case, let [tex]\( x = a n + b \)[/tex] and [tex]\( y = a h \)[/tex]:
[tex]\[
(a n + a h + b)^5 = (a n + b + a h)^5
\][/tex]
Expanding this, we get:
[tex]\[
(a n + b)^5 + 5 (a n + b)^4 (a h) + 10 (a n + b)^3 (a h)^2 + 10 (a n + b)^2 (a h)^3 + 5 (a n + b) (a h)^4 + (a h)^5
\][/tex]
5. Subtract [tex]\( (a n + b)^5 \)[/tex]:
[tex]\[
[(a n + b)^5 + 5 (a n + b)^4 (a h) + 10 (a n + b)^3 (a h)^2 + 10 (a n + b)^2 (a h)^3 + 5 (a n + b) (a h)^4 + (a h)^5] - (a n + b)^5
\][/tex]
This simplifies to:
[tex]\[
5 (a n + b)^4 (a h) + 10 (a n + b)^3 (a h)^2 + 10 (a n + b)^2 (a h)^3 + 5 (a n + b) (a h)^4 + (a h)^5
\][/tex]
6. Divide by [tex]\( h \)[/tex] and simplify:
[tex]\[
\frac{5 (a n + b)^4 (a h) + 10 (a n + b)^3 (a h)^2 + 10 (a n + b)^2 (a h)^3 + 5 (a n + b) (a h)^4 + (a h)^5}{h}
\][/tex]
This can be simplified to:
[tex]\[
5 (a n + b)^4 a + 10 (a n + b)^3 (a h) + 10 (a n + b)^2 (a h)^2 + 5 (a n + b) (a h)^3 + (a h)^4
\][/tex]
7. Take the limit as [tex]\( h \to 0 \)[/tex]:
[tex]\[
\lim_{h \to 0} \left[ 5 (a n + b)^4 a + 10 (a n + b)^3 (a h) + 10 (a n + b)^2 (a h)^2 + 5 (a n + b) (a h)^3 + (a h)^4 \right]
\][/tex]
As [tex]\( h \to 0 \)[/tex], terms containing [tex]\( h \)[/tex] will vanish:
[tex]\[
5 (a n + b)^4 a
\][/tex]
Thus, the derivative of the function [tex]\( f(n) = (a n + b)^5 \)[/tex] from first principles is:
[tex]\[
f'(n) = 5 a (a n + b)^4
\][/tex]
