SME
Calculus I · Problem type
The derivative of A·eᵏˣ
Use this for “Find the slope of y = 3e0.5x at x = 2.”
Chart
f(3) ≈ 2.744
Tangent at x = 3: slope f′(3) = −0.5488, y = −0.5488x + 4.39
This is an example. Type your own values to start a new problem.
f(x)
=
A
e
x
f′(
x
)
=
f′(x)
f(x)
Value there
calculated
Formulas
f
(
x
)
=
A
×
e
k
×
x
2.7441
=
5
×
e
−0.2 × 3
f
′(
x
)
=
k
×
f
(
x
)
−0.5488 = −0.2 × 2.7441
Assumptions
•
The slope of eˣ is eˣ itself. For A·eᵏˣ the inside kx has slope k (chain rule), so f′(x) = k·A·eᵏˣ = k·f(x).
•
With k > 0 the curve moves away from 0 faster and faster and its slope has the sign of A; with k < 0 it decays toward 0 and its slope has the opposite sign.
•
ln x, the inverse of eˣ, has slope 1 ÷ x for x > 0.
Step-by-step
WE KNOW
A = 5, k = −0.2, x = 3
FIND
value there (f(x)), slope of the tangent (f′(x))
Step 1 · Find value there (f(x))
Use f(x) = A × ek × x
Raise
e
to
the
power
k
times
x
,
then
multiply
by
A
.
f
(
x
)
=
5
×
e
−0.2 × 3
f
(
x
)
=
5
×
e
−0.6
f
(
x
)
=
5
×
0.5488
f
(
x
)
=
2.7441
Step 2 · Find slope of the tangent (f′(x))
Use f′(x) = k × f(x)
Chain
rule:
the
slope
of
e
u
is
e
u
,
times
the
inside’s
slope
k
.
So
the
slope
is
k
times
the
value.
f
′(
x
)
=
−0.2
×
2.7441
y(
x
)
=
5
e
−0.2
x
→
dy/dx
=
−
e
−0.2
x
dy/dx
at
x
=
3
=
−
e
−0.6
=
−0.5488
f
′(
x
)
=
−0.5488
Check
Put the numbers back in. Both sides should match.
2.7441
=
5
×
e
−0.2 × 3
✓
−0.5488 = −0.2 × 2.7441 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Derivatives and differentiation rules
Main lesson
The product and quotient rules from a table
Use this for “f(2) = 3, f′(2) = −1, g(2) = 4, g′(2) = 5. Find (fg)′(2) and (f/g)′(2).”
The chain rule on a power
Use this for “Find f′(1) for f(x) = (2x + 1)³.”
The derivative of A sin(bx)
Use this for “Find the slope of y = 4 sin(2x) at x = π/3.”
Implicit slope on a circle
Use this for “Find dy/dx on x² + y² = 100 at (6, −8), and the tangent line there.”
Linear approximation of √x
Use this for “Use the linearization of √x at a = 25 to estimate √26.”
L’Hôpital’s rule on (ekx − 1) ÷ (mx)
Use this for “Find the limit of (e4x − 1) ÷ (5x) as x → 0.”