SME
Calculus I · Problem type
The derivative of A sin(bx)
Use this for “Find the slope of y = 4 sin(2x) at x = π/3.”
Chart
f(π/6) = 2
Tangent at x = π/6: slope f′(π/6) = 0, y = 2
This is an example. Type your own values to start a new problem.
f(x)
=
A
sin(
b
x)
f′(
x
)
=
f′(x)
bx
Angle inside the sine
worked out (not typed)
f(x)
Value there
calculated
Formulas
bx
=
b
×
x
π
2
=
3
×
π
6
f
(
x
)
=
A
×
sin(
bx
)
2
=
2
×
sin(
π
2
)
f
′(
x
)
=
A
×
b
×
cos(
bx
)
0
=
2
×
3
×
cos(
π
2
)
Assumptions
•
x is in radians; the slope of sin x is cos x only when x is measured in radians.
•
The slope of sin x is cos x, and the slope of cos x is −sin x.
•
For A sin(bx) the inside bx has slope b (chain rule), so f′(x) = A·b·cos(bx).
•
Where cos(bx) = 0 the tangent is flat: a top or a bottom of the wave.
Step-by-step
WE KNOW
A = 2, b = 3, x = π/6
FIND
angle inside the sine (bx), value there (f(x)), slope of the tangent (f′(x))
Step 1 · Find angle inside the sine (bx)
Use bx = b × x
Work out the inside first: the angle the sine is taken of.
bx
=
3
×
π
6
bx
=
π
2
(≈
1.57)
Step 2 · Find value there (f(x))
Use f(x) = A × sin(bx)
Take
the
sine
of
the
angle,
then
multiply
by
A
.
f
(
x
)
=
2
×
sin(
π
2
)
f
(
x
)
=
2
Step 3 · Find slope of the tangent (f′(x))
Use f′(x) = A × b × cos(bx)
Chain
rule:
sin
turns
into
cos
at
the
same
angle,
times
the
inside’s
slope
b
,
with
A
in
front,
so
f
′(
x
)
=
6
cos(
3
x
).
f
′(
x
)
=
2
×
3
×
cos(
π
2
)
f
′(
x
)
=
6
×
0
f
′(
x
)
=
0
Check
Put the numbers back in. Both sides should match.
π
2
=
3
×
π
6
✓
2
=
2
×
sin(
π
2
)
✓
0
=
2
×
3
×
cos(
π
2
)
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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Use this for “Find the limit of (e4x − 1) ÷ (5x) as x → 0.”