Theorem (Chain Rule): If `g` is differentiable at `a` and `f` is differentiable at `b=f(a)`, and `h(x) = f(g(x))` for an interval `I` with `a in I`
then `h` is differentiable at `a` and `h'(a) = f'(g(a)) cdot g'(a)`.
PROOF of Chain Rule:
Part I: Assume there is some interval ,`I` containing `a` where for all `x in I` , `g(x) ne g(a)`.![Figure 1.](data:image/png;base64,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)
Let `b = g(a)`and `k = g(a+h) - g(a) = g(x) - g(a)` for `h ne 0`.
Note `g(x) = g(a+h) = g(a) + k = b + k`. See Figure 1 .
From the assumption that `g` is differentiable at `a`, we have that `g`
is also continuous at `a` . Thus we can conclude that as `h to 0, k to
0`.
We’ll follow the usual steps in finding the derivative of P at a:
Step I: `P(a+h) = f(g(a+h))`
`\underline{- P(a)\ \ \ \ = f(g(a))}`
Step II: `P(a+h) - P(a) = f(g(a+h)) - f(g(a)) = f(b + k) - f(b)`.
Now we assumed that `k ne 0` . [Note: This is a major assumption for some functions.]
So
`P(a+h) - P(a) = {f(b + k) - f(b)}/k . k`
Therefore
`P'(a) = lim_{h to 0} { P(a+h) - P(a)}/h`
`= lim _{h to 0, k to 0} {f(b + k) - f(b)}/k cdot k/h`
`= lim _{h to 0, k to 0} {f(b + k) - f(b)}/k cdot {g(a+h)-g(a) }/h`
`= f '(b) cdot g'(a)`
`= f '(g(a)) cdot g'(a)`.
Part II: Recall that we had `k = g(a+h)-g(a)` for `h ne 0`.
Suppose that `k = 0` for values of `h` arbitrarily close to `0`.
Since we assume that g is differentiable we know that![](data:image/png;base64,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)
`lim_{h to 0} {g(a+h)-g(a)}/ h` must exist. Our assumption that `k = 0` for `h` arbitrarily close to `0` means that
there is a sequence of values of ` h`, {`h_n`} with `h_n to 0` and `g(a+h_n) -g(a) = 0` for all `n`.
Thus `lim_{n to oo} {g(a+h_n)-g(a)}/ {h_n } = lim_{n to oo} 0/ {h_n }= 0` . [See Figure 2 ]
Thus `g'(a) = lim_{ h to 0} {g(a+h)-g(a)} / h = 0`. [0 is the only possible limit.]
To complete the argument we need only show that `P'(a)= 0`.
But for precisely the same h values that had `k = g(a+h)-g(a) = 0`, we have `g(a+h) = g(a)`. Thus for these values of `h`
`P(a+h) - P(a) = f(g(a+h)) - f(g(a)) = f(b + k) - f(b) = f(b) - f(b) = 0.`![](data:image/png;base64,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)
and hence `{P(a+h) - P(a)}/h = 0`. [See Figure 3]
Now for any `h` where `k ne 0`, see Figure 1, the argument of part
I is still valid to show that `{P(a+h) - P(a)}/h to 0` as `h to
0`.
[This is primarily because `g'(a)=0`.]
In summary then , as h approaches 0 either `{P(a+h)-P(a)}/h` is
close to or actually is 0.
Thus `P'(a) =lim_{h to 0} {P(a+h) - P(a) }/h= 0`. EOP.