A-Level Topic: Differentiating Logarithms 8 Starter and ...
A-Level Starter Activity
Topic: Differentiating Logarithms
and Exponentials
Chapter Reference: Pure 2, Chapter 9
8 minutes
1. Find the value of f'(x) when,
a. f(x) = 3x + ex
(1)
1
b. f(x) = 2 + 2 ln
(1)
c.
f(x)
=
4
+
1 4
ln
(1)
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2.
Find
the
value
of
x
for
which
f'(x)
=
-1
when
f(x)
=
2 8
-
2
+
ln
(3)
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3. Find the equation for the normal to the curve y = 4 + 3ex when x = 0
(4)
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1
4. Find he equation for the tangent to the curve y = 3 - 3 at the point when x = 1.
(4)
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Solutions
1a.
f'(x) = 3 + ex
M1
1b.
f'(x)
=
1 2
-12
+
2
M1
1c.
f'(x)
=
2 -12
+
1 4
M1
2.
f'(x)
=
1 4
?
2
+
1
1 4
?
2
+
1
=
-1
M1
x2 ? 4x + 4 = 0 (x ? 2)2 = 0
M1
x = 2
M1
3.
When x = 0, y = 7
M1
=
3x
M1
Gradient = 3
Gradient of normal = -1
3
M1
y
=
7
-
1 3
M1
4.
x = 1 y = 1 ? 3e
M1
=
1 3
-23
-
3
Gradient
=
1 3
-
3
M1
y
?
(1
?
3e)
=
(1
3
-
3)(
-
1)
M1
y
=
(13
-
3)
+
2 3
M1
A-Level Starter Activity
Topic: Differentiating sin and cos 8
Chapter Reference: Pure 2, Chapter 9
minutes
1. Differentiate:
a. y = cos (3x ? 2)
(1)
b.
y
=
4
sin
(
3
-
)
(1)
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2. Find the coordinate of any stationary points on the curve y = x + 2 sin x in the interval 0 x 2
(4)
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3. Find the equation for the tangent to the curve y = cos x at the point x =
3
(5)
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Solutions
1a.
=
-3 sin(3 - 2)
M1
1b.
=
-4
cos
(3
-
x)
M1
2.
=
1
+
2
cos
M1
S.P: 1 + 2 cos x = 0
cos x = -12
M1
x
=
-
3
=
2 3
y
=
2 3
+
3
M1
x
=
+
3
=
4 3
y
=
4 3
-
3
M1
3.
=
3
y = 1
M1
2
=
- sin
M1
gradient = -sin (3) = -23
M1
y
-
1 2
=
-
3 2
(
-
)
3
M1
33 + 6y ? 3 - 3 = 0
M1
A-Level Starter Activity
Topic: Differentiating tan, sec,
cosec, and cot
Chapter Reference: Pure 2, Chapter 9
7 minutes
1. Differentiate 3 cosec (x + )
6
(1)
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2. Differentiate sec22x
(1)
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3. Differentiate ln (tan 4x)
(2)
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4. Differentiate 2 cot x2
(1)
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5. Find the equation for the tangent to the curve y = cosec x ? 2 sin x at the point x =
6
(4)
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