- 2,020
- 843
I'm having some problems using the chain rule and I'm not sure where the trouble lies. For example:
If I'm not mistaken, if we have the composite function f(g(n)) then [math]\Delta f(g(n)) = \dfrac{ \Delta f(g) }{ \Delta g } \dfrac{ \Delta g(n) }{ \Delta n }[/math]
Let [math]f(g(n)) = (n^2)^2[/math]. Then [math]f(g) = g^2[/math] and [math]g(n) = n^2[/math]
[math]\Delta f(g(n)) = \dfrac{ \Delta g^2 }{ \Delta g } \dfrac{ \Delta n^2 }{ \Delta n }[/math]
Now, [math]\Delta g^2 = (g + 1)^2 - g^2 = 2g + 1[/math]. (And similarly for [math]\Delta n^2[/math].)
So
[math]\Delta f(g(n)) = (2g + 1)(2n + 1) = (2(n^2) + 1)(2n + 1) = 4 n^3 + 2 n^2 + 2 n + 1[/math]
But if we calculate [math]\Delta n^4[/math] from the definition:
[math]\Delta n^4 = (n + 1)^4 - n^4 = 4 n^3 + 6 n^2 + 4 n + 1[/math]
Where am I going wrong?
-Dan
If I'm not mistaken, if we have the composite function f(g(n)) then [math]\Delta f(g(n)) = \dfrac{ \Delta f(g) }{ \Delta g } \dfrac{ \Delta g(n) }{ \Delta n }[/math]
Let [math]f(g(n)) = (n^2)^2[/math]. Then [math]f(g) = g^2[/math] and [math]g(n) = n^2[/math]
[math]\Delta f(g(n)) = \dfrac{ \Delta g^2 }{ \Delta g } \dfrac{ \Delta n^2 }{ \Delta n }[/math]
Now, [math]\Delta g^2 = (g + 1)^2 - g^2 = 2g + 1[/math]. (And similarly for [math]\Delta n^2[/math].)
So
[math]\Delta f(g(n)) = (2g + 1)(2n + 1) = (2(n^2) + 1)(2n + 1) = 4 n^3 + 2 n^2 + 2 n + 1[/math]
But if we calculate [math]\Delta n^4[/math] from the definition:
[math]\Delta n^4 = (n + 1)^4 - n^4 = 4 n^3 + 6 n^2 + 4 n + 1[/math]
Where am I going wrong?
-Dan
Last edited by a moderator: