1.4. THE DERIVATIVE FUNCTION
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We now may take two different perspectives on thinking about the derivative function:
given a graph of y = f (x), how does this graph lead to the graph of the derivative
function y = f ′ (x)? and given a formula for y = f (x), how does the limit definition of
the derivative generate a formula for y = f ′ (x)? Both of these issues are explored in the
following activities.
Activity 1.10.
For each given graph of y = f (x), sketch an approximate graph of its derivative
function, y = f ′ (x), on the axes immediately below. The scale of the grid for the graph
of f is 1 × 1; assume the horizontal scale of the grid for the graph of f ′ is identical to
that for f . If necessary, adjust and label the vertical scale on the axes for f ′ .
f
x
f ′
x
g
x
g ′
x
p
x
p ′
x
q
x
q ′
x
37
We now may take two different perspectives on thinking about the derivative function:
given a graph of y = f (x), how does this graph lead to the graph of the derivative
function y = f ′ (x)? and given a formula for y = f (x), how does the limit definition of
the derivative generate a formula for y = f ′ (x)? Both of these issues are explored in the
following activities.
Activity 1.10.
For each given graph of y = f (x), sketch an approximate graph of its derivative
function, y = f ′ (x), on the axes immediately below. The scale of the grid for the graph
of f is 1 × 1; assume the horizontal scale of the grid for the graph of f ′ is identical to
that for f . If necessary, adjust and label the vertical scale on the axes for f ′ .
f
x
f ′
x
g
x
g ′
x
p
x
p ′
x
q
x
q ′
x