# 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, $(f_1 \circ f_2 \circ f_3)(x) := f_1(f_2(f_3(x)))$.

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