SME
Calculus I
Derivatives and differentiation rules
Use this for “Find the slope of y = 3x⁴ at x = −1.”
Chart
f(1.5) = 2.25
Tangent at x = 1.5: slope f′(1.5) = 3, y = 3x − 2.25
This is an example. Type your own values to start a new problem.
f(x)
=
c
x
f′(
x
)
=
f′(x)
f(x)
Value there
calculated
Formulas
f
(
x
)
=
c
×
x
n
2.25
=
1
×
1.5
2
f
′(
x
)
=
n
×
c
×
x
n
− 1
3
=
2
×
1
×
1.5
2 − 1
Assumptions
•
f′(x) = lim (h → 0) [f(x + h) − f(x)] ÷ h: the slope of the tangent line at x.
•
Power rule: the derivative of xⁿ is n·xⁿ⁻¹. Constant-multiple rule: c stays in front.
•
f(x) = c·xⁿ with n a whole number 0–6. With n = 0, f is the constant c and its slope is 0.
Step-by-step
WE KNOW
x = 1.5, c = 1, n = 2
FIND
value there (f(x)), slope of the tangent (f′(x))
Step 1 · Find value there (f(x))
Use f(x) = c × xⁿ
Raise
x
to
the
power
n
,
then
multiply
by
c
.
f
(
x
)
=
1
×
1.5
2
f
(
x
)
=
2.25
Step 2 · Find slope of the tangent (f′(x))
Use f′(x) = n × c × xⁿ⁻¹
Power
rule:
bring
n
down
in
front
and
lower
the
power
by
1.
Constant-multiple
rule:
keep
c
.
f
′(
x
)
=
2
×
1
×
1.5
2 − 1
y(
x
)
=
x
2
→
dy/dx
=
2
x
dy/dx
at
x
=
1.5
=
2(1.5)
=
3
f
′(
x
)
=
3
Check
Put the numbers back in. Both sides should match.
2.25
=
1
×
1.5
2
✓
3
=
2
×
1
×
1.5
2 − 1
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
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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.”
The derivative of A·eᵏˣ
Use this for “Find the slope of y = 3e0.5x at x = 2.”
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.”