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The Chain Rule
You probably remember the derivatives of sin(x), x8, and ex.
But what about functions like sin(2x-1), (3x2-4x+1)8, or e-x2?
How do we take the derivative of compositions of functions?
The Chain Rule allows us to use our knowledge of
the derivatives of functions f(x) and g(x) to find
the derivative of the composition f(g(x)):
Suppose a function g(x) is differentiable at x and f(x)
is differentiable at g(x). Then the composition f(g(x))
is differentiable at x.
Letting y = f(g(x)) and u = g(x),
dy dy du
─=─*─
dx du dx
Using alternative notation,
d
─﹝f(g(x))﹞=f'(g(x))g'(x),
dx
d.............................du
─﹝f(u)﹞=f'(u) ─
dx...........................dx
You probably remember the derivatives of sin(x), x8, and ex.
But what about functions like sin(2x-1), (3x2-4x+1)8, or e-x2?
How do we take the derivative of compositions of functions?
The Chain Rule allows us to use our knowledge of
the derivatives of functions f(x) and g(x) to find
the derivative of the composition f(g(x)):
Suppose a function g(x) is differentiable at x and f(x)
is differentiable at g(x). Then the composition f(g(x))
is differentiable at x.
Letting y = f(g(x)) and u = g(x),
dy dy du
─=─*─
dx du dx
Using alternative notation,
d
─﹝f(g(x))﹞=f'(g(x))g'(x),
dx
d.............................du
─﹝f(u)﹞=f'(u) ─
dx...........................dx
