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Summary: The general procedure in calculating derivatives using the chain
rule
includes the following
steps:
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Step 1: Identify
the dependent variable, y, and independent variable x.
i.e. y
= y(x) =
e sin x
Here y
is the dependent variable
and x is the independent variable
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Step 2: Select a
convenient intermediate variable, u.
i.e. u
= sin x Therefore du/dx = cos x ( a
“chain link” to be used later)
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Step 3: Write
the dependent variable in terms of the intermediate variable.
i.e. y
= y [ u(x) ] = e
u , Therefore dy/du = e u ( a “chain link” to be used later)
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Step 4. Multiply
the “chain links” together to obtain the derivative of y(x)
using the “chain rule”
i.e. dy/dx = dy/du du/dx =
e u cos x
↑ ↑
chain
link chain link
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Step 5. Express
result in terms of the independent variable, x.
i.e. dy/dx = e sin x cos x (result)
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Click here for examples involving the chain rule.
Calculating higher order
derivatives using the chain rule is slightly more complicated.
Click here for a
discussion of the procedure to take second order derivatives using
the chain rule.
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