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Calculus · SEC2_GYM
Derivative as local rate of change
Foundation
Core
Stretch
Question 1/8
+12 XP
Differentiate
f
(
x
)
=
3
x
3
.
Secant becomes tangent
f(x) = 3x³, x = 1
0
0.5
1
1.5
5
10
15
x
f(x)
Δx
Δy
(
1
,
3
)
slopes
f
(
x
)
=
3
x
3
Δx
=
0.6
→
m
=
15.48
Δx
=
0.3
→
m
=
11.97
Δx
=
0.1
→
m
=
9.93
Δy
Δx
=
f
′
(
1
)
=
9
curve f
secant Δx = 0.6
secant Δx = 0.3
secant Δx = 0.1
tangent, slope 9
point of tangency
All three secants pass through (1, 3). As Δx shrinks (0.6 → 0.3 → 0.1) the secant slope moves 15.48 → 11.97 → 9.93 — towards the tangent slope f′(1) = 9. That limit is the derivative.
Your answer
Hint 1/3
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