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Mathematics A
1. Pure Mathematics
1.7 Differentiation
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Differentiation determines the rate of change of a
function
Differentiation is used to find the
gradient
of a curve at any point.
The outcome of differentiation is called the
derivative
Differentiation is used to find maximum and minimum values of a
function
.
Differentiation calculates the rate of change of a
function
Integration finds the area under a
curve
.
The general power rule for differentiation states that if
y
=
y =
y
=
x
n
x^{n}
x
n
, then
d
y
/
d
x
=
dy / dx =
d
y
/
d
x
=
n
x
n
−
1
nx^{n - 1}
n
x
n
−
1
. This rule is foundational for derivatives of polynomials
The derivative of a
constant
is always 0.
The product rule for differentiation is d / dx (uv) = u'v + uv'</latex>, where
u
′
u'
u
′
and
v
′
v'
v
′
represent the derivatives
Match the differentiation rule with its example:
Product Rule ↔️
d
/
d
x
(
x
2
s
i
n
(
x
)
)
=
d / dx(x^{2}sin(x)) =
d
/
d
x
(
x
2
s
in
(
x
))
=
2
x
s
i
n
(
x
)
+
2x sin(x) +
2
x
s
in
(
x
)
+
x
2
c
o
s
(
x
)
x^{2} cos(x)
x
2
cos
(
x
)
Quotient Rule ↔️
d
/
d
x
(
x
/
(
x
+
1
)
)
=
d / dx(x / (x + 1)) =
d
/
d
x
(
x
/
(
x
+
1
))
=
1
/
(
x
+
1
)
2
1 / (x + 1)^{2}
1/
(
x
+
1
)
2
Chain Rule ↔️
d
/
d
x
(
(
x
+
1
)
2
)
=
d / dx((x + 1)^{2}) =
d
/
d
x
((
x
+
1
)
2
)
=
2
(
x
+
1
)
2(x + 1)
2
(
x
+
1
)
The derivative of
e
x
e^{x}
e
x
is
e
x
e^{x}
e
x
.
The derivative of
l
n
(
x
)
ln(x)
l
n
(
x
)
is
1
/
x
1 / x
1/
x
, which is valid for positive values of
x
x
x
.
Order the derivatives of common trigonometric functions:
1️⃣
s
i
n
(
x
)
−
>
c
o
s
(
x
)
sin(x) - > cos(x)
s
in
(
x
)
−
>
cos
(
x
)
2️⃣
c
o
s
(
x
)
−
>
−
s
i
n
(
x
)
cos(x) - > - sin(x)
cos
(
x
)
−
>
−
s
in
(
x
)
What is the derivative of
l
n
(
x
)
ln(x)
l
n
(
x
)
?
1
/
x
1 / x
1/
x
Differentiation calculates the rate of change of a function, also known as the
gradient
Differentiation is used to find maximum and minimum values of a
function
.
Match the concept with its definition or use case:
Differentiation ↔️ Calculates the rate of change of a function
Integration ↔️ Reverses differentiation and finds the area under a curve
What does the power rule for differentiation state if
y
=
y =
y
=
x
n
x^{n}
x
n
?
d
y
d
x
=
\frac{dy}{dx} =
d
x
d
y
=
n
x
n
−
1
nx^{n - 1}
n
x
n
−
1
According to the power rule, the derivative of
x
3
x^{3}
x
3
is
3
x
2
3x^{2}
3
x
2
The derivative of a
constant
is always zero.
What is the derivative of
a
x
+
ax +
a
x
+
b
b
b
?
a
a
a
What does the power rule for differentiation state if
y
=
y =
y
=
x
n
x^{n}
x
n
?
d
y
d
x
=
\frac{dy}{dx} =
d
x
d
y
=
n
x
n
−
1
nx^{n - 1}
n
x
n
−
1
The derivative of a constant is always
zero
The power rule for differentiation is foundational for
polynomials
.
Match the function with its derivative:
x
n
x^{n}
x
n
↔️
n
x
n
−
1
nx^{n - 1}
n
x
n
−
1
c
c
c
↔️
0
0
0
a
x
+
ax +
a
x
+
b
b
b
↔️
a
a
a
What is the formula for the product rule of differentiation?
\frac{d}{dx}(uv) = u'v + uv'</latex>
The quotient rule states that
d
d
x
(
u
v
)
=
\frac{d}{dx}(\frac{u}{v}) =
d
x
d
(
v
u
)
=
u
′
v
−
u
v
′
v
2
\frac{u'v - uv'}{v^{2}}
v
2
u
′
v
−
u
v
′
, where the denominator is v^{2}
Steps to apply the chain rule
1️⃣ Identify the outer and inner functions
2️⃣ Differentiate the outer function
3️⃣ Differentiate the inner function
4️⃣ Multiply the results
What is the derivative of
s
i
n
(
x
)
sin(x)
s
in
(
x
)
?
c
o
s
(
x
)
cos(x)
cos
(
x
)
The derivative of
c
o
s
(
x
)
cos(x)
cos
(
x
)
is -sin(x)</latex>
What is the derivative of
e
x
e^{x}
e
x
?
e
x
e^{x}
e
x
The derivatives of
s
i
n
(
x
)
sin(x)
s
in
(
x
)
and
c
o
s
(
x
)
cos(x)
cos
(
x
)
oscillate between each other.
What does the derivative function represent in terms of a curve?
The slope of the curve
For
y
=
y =
y
=
3
x
2
+
3x^{2} +
3
x
2
+
2
x
−
1
2x - 1
2
x
−
1
, the derivative is \frac{dy}{dx} = 6x + 2
Steps to find the gradient of a curve at a point
1️⃣ Differentiate the function
2️⃣ Substitute the x value into the derivative
3️⃣ Calculate the gradient