Quiz:Chain rule for differentiation

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See chain rule for differentiation and chain rule for higher derivatives for background information.

See Quiz:Differentiation rules for a quiz on all the differentiation rules together.

Formulas

1 Suppose f and g are both twice differentiable functions everywhere on \R. Which of the following is the correct formula for (f \circ g)'', the second derivative of the composite of two functions?

(f'' \circ g) \cdot g''
(f'' \circ g) \cdot (f' \circ g') \cdot g''
(f'' \circ g) \cdot (f' \circ g') \cdot (f \circ g'')
(f'' \circ g) \cdot (g')^2 + (f' \circ g) \cdot g''
(f' \circ g') \cdot (f \circ g) + (f'' \circ g'')

2 Suppose f_1,f_2,f_3 are everywhere differentiable functions from \R to \R. What is the derivative (f_1 \circ f_2 \circ f_3)' where \circ denotes the composite of two functions? In other words, (f_1 \circ f_2 \circ f_3)(x) := f_1(f_2(f_3(x))).

(f_1' \circ f_2 \circ f_3) \cdot (f_2' \circ f_3) \cdot f_3'
(f_1' \cdot f_2 \cdot f_3) \circ (f_2' \cdot f_3) \circ f_3'
(f_1 \circ f_2' \circ f_3') \cdot (f_2 \circ f_3') \cdot f_3
(f_1 \cdot f_2' \cdot f_3') \circ (f_2 \cdot f_3') \circ f_3
f_1' \circ f_2' \circ f_3'