The Chain Rule
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The Chain Rule
If the function f has derivative f' and the function u has derivative du/dx, then the composite function f(u) is differentiable, and
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The Chain Rule
If the function f has derivative f' and the function u has derivative du/dx, then the composite function f(u) is differentiable, and
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Proof From the definition of the derivative,
 h |
 u'(x) as h0. |
Thus,
h |
- | u'(x) | 0 as h0. |
If we take v to be the quantity
| v | = | h |
- | u'(x) | , |
then v0 as h0. If we solve this equation for u(x+h), we get
where v0 as h0
Now the same is true for f.
where w0 as h0.
What we are after is the derivative of f(u(x)). Thus we need to calculate the limit of
h |
First look at the numerator: f(u(x+h)) - f(u(x)). If we use formula (I) to substitute for u(x+h) we get
Now we use formula (II) to rewrite f [u(x) + (u'(x) + v)h]:
Subtracting f(u(x)) from both sides gives
Putting (III) and (IV) together now gives
This is the numerator of the expression we are after. Dividing by h gives
h |
= | h |
= | (f'(u(x)) + w)(u'(x) + v). |
Now let h0. Since both v and w0, we obtain
| lim h 0 |
h |
= | f'(u(x)) u'(x), |
which is the chain rule.
