Differentiation Pt. 3: General Differentiation 1. Find 2 2 for y = 4x 3 + e x (2) 2. Find for y = 3 + 3 ln (2) 3. Find any values of x for which = 0 when y = 5ln x โ 8x (2) 4. Find the co-ordinates and the nature of any stationary points on the curve y = 7 + 2x โ 4 lnx (5) 5. Find an equation for the normal to the cube y = 10 + ln x at the x-cordinate 3. (4) 6. Find the differential of 4sin x + 2 cos x (2) 7. Differentiate cot(2x โ 3) (1) 8. Find an equation for the tangent for the curve y = cosec x โ 2sin x at the point x = 6 in the interval 0 โค x โค 2(5) 9. Find the differential of y = sec 5x (1) 10. Differentiate with respect to x, 3 x 11. Differentiate with respect to x, 5 1-x 12. Differentiate with respect to x, 2 3 A-Level Pt. 3: General Differentiation Pt. 4: The Chain Rule Pt. 5: The Product Rule Pt. 6: The Quotient Rule Pt. 7: Parametric Differentiation Pt. 8: Implicit Differentiation Pt. 9: Rates of Change AS Level Pt. 1: First Principles Pt. 2: Differentiation
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AS Level -Level Differentiation Pt. 3: General Differentiation...Differentiation Pt. 5: Product Rule 1. Differentiate xex with respect to x. (3) 2. Find ๐ ๐ when y = x2 ln (x
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Differentiation
Pt. 3: General Differentiation
1. Find ๐2๐ฆ
๐๐ฅ2 for y = 4x3 + ex (2)
2. Find ๐๐ฆ
๐๐ฅ for y =
3
๐ฅ+ 3 ln ๐ฅ (2)
3. Find any values of x for which ๐๐ฆ
๐๐ฅ = 0 when y = 5ln x โ 8x (2)
4. Find the co-ordinates and the nature of any stationary points on the curve y = 7 + 2x โ 4 lnx (5)
5. Find an equation for the normal to the cube y = 10 + ln x at the x-cordinate 3. (4)
6. Find the differential of 4sin x + 2 cos x (2)
7. Differentiate cot(2x โ 3) (1)
8. Find an equation for the tangent for the curve y = cosec x โ 2sin x at the point x = ๐
6 in the interval 0 โค x โค 2๐ (5)
9. Find the differential of y = sec 5x (1)
10. Differentiate with respect to x, 3x
11. Differentiate with respect to x, 51-x
12. Differentiate with respect to x, 2๐ฅ3
A-Level
Pt. 3: General Differentiation
Pt. 4: The Chain Rule
Pt. 5: The Product Rule
Pt. 6: The Quotient Rule
Pt. 7: Parametric Differentiation
Pt. 8: Implicit Differentiation
Pt. 9: Rates of Change
AS Level
Pt. 1: First Principles
Pt. 2: Differentiation
Pt. 9: Rates of Change
Pt. 1: First Principles
Pt. 2: Differentiation
Mark Scheme
1.
๐๐ฆ
๐๐ฅ= 12x2 + ex M1
๐2๐ฆ
๐๐ฅ2 = 24x + ex M1
2.
y = 3
๐ฅ+ 3 ln ๐ฅ = 3x-1 + 3 ln x M1
๐๐ฆ
๐๐ฅ = -3x-2 +
3
๐ฅ M1
3.
๐๐ฆ
๐๐ฅ =
5
๐ฅ - 8 M1
5
๐ฅ โ 8 = 0
5
๐ฅ = 8
x = 5
8
M1
3.
๐๐ฆ
๐๐ฅ = 2 -
4
๐ฅ M1
Stationary point: 2 - 4
๐ฅ = 0
x = 2 M1
๐2๐ฆ
๐๐ฅ2 = 4x-2 M1
x = 2, ๐2๐ฆ
๐๐ฅ2 = 1
๐2๐ฆ
๐๐ฅ2 > 0, therefore minimum point M1
x = 2, y = 11 โ 4ln 2 M1
Therefore (2, 11 โ 4ln 2) is a minimum point
5.
x = 3, y = 10 + ln 3 M1
๐๐ฆ
๐๐ฅ=
1
๐ฅ M1
When x = 3, ๐๐ฆ
๐๐ฅ =
1
3
Gradient of normal = -3 M1
Therefore equation of line is:
y โ (10 + ln 3) = -3(x โ 3)
y = 19 + ln 3 โ 3x
M1
6.
๐๐ฆ
๐๐ฅ= 4cos x โ 2 sin x M1
7.
๐๐ฆ
๐๐ฅ= -2coscec2(2x โ 3) M1
8.
๐โ๐๐ ๐ฅ =๐
6, ๐ฆ = 1 M1
๐๐ฆ
๐๐ฅ = - cosec x cot x โ 2 cos x M1
๐๐ฆ
๐๐ฅ (
๐
6) = โ cosec (
๐
6) cot (
๐
6) โ 2 cos (
๐
6)
๐๐ฆ
๐๐ฅ = 3โ3
M1
y โ 1 = 3โ3 (x - ๐
6) M1
6โ3๐ฅ + 2y โ 2 - โ3๐ = 0 M1
9.
๐๐ฆ
๐๐ฅ = 5 sec 5x tan 5x M1
10.
๐
๐๐ฅ(3x) = 3x ln 3 M1
11.
๐
๐๐ฅ(51 โ x) = 51 โ x ln 5 ร -1 M1
๐
๐๐ฅ(51 โ x) = -(51 โ x)ln 5
12.
๐
๐๐ฅ(2๐ฅ3
) = 2๐ฅ3ln 2 ร 3x2 M1
๐
๐๐ฅ(2๐ฅ3
) = 3x2(2๐ฅ3) ln 2
Differentiation
Pt. 4: Chain Rule
1. Differentiate with respect to x, (x + 3)5 (1)
2. Find ๐2๐ฆ
๐๐ฅ2 when y = 4 ln (1 + 2x) (2)
3. Find the value of fโ(x) when f(x) = 4โ3๐ฅ + 15 (2)
4. Find the coordinates of any stationary points for the curve y = 8x โ e2x (4)
5. Find an equation for the tangent to the curve y = 2 + ln (1 + 4x) given the x-coorinate is 0. (4)
6. Find an equation for the normal to the curve y = 1
2โln ๐ฅ when x = 1. (5)
7. Find the coordinates of any stationary points on the curve y = 2 ln(x โ x2). (5)
8. A curve has the equation y = โ8 โ ๐2๐ฅ
The point P on the curve has y-coordinate 2.
a. Find the x-coordinate of P. (2)
b. Show that the tangent to the curve at P has equation 2x + y = 2 + ln 4. (2)
9. A quantity N is increasing such that at time t seconds, N = aekt.
Given that at time t = 0, N = 20 and that at time t = 8, N = 60, find
a. The values of the constants a and k (3)
b. The value of N when t = 12 (2)
c. The rate at which N is increasing when t = 12. (3)
10. f(x) = (5 โ 2x2)3
a. Find fโ(x) (2)
b. Find the coordinates of the stationary points of the curve y = f(x). (4)
c. Find the equation for the tangent to the curve y = f(x) at the point with x-coordinate 3
2 giving your answer in the
form ax + by + c = 0, where a, b and c are integers. (3)
11. The diagram shows the curve y = f(x) where f(x) = 3 ln 5x โ 2x, x > 0.
a. Find f โฒ(x). (1)
b. Find the x-coordinate of the point on the curve at which the gradient of the normal to the curve is -1
4 . (2)
c. Find the coordinates of the maximum turning point of the curve. (3)
d. Write down the set of values of x for which f(x) is a decreasing function. (1)
12. f(x) โก x2 โ 7x + 4 ln (๐ฅ
2), x > 0.
a. Solve the equation f โฒ(x) = 0, giving your answers correct to 2 decimal places. (3)
b. Find an equation for the tangent to the curve y = f(x) at the point on the curve where x = 2. (2)
A-Level
Pt. 3: General Differentiation
Pt. 4: The Chain Rule
Pt. 5: The Product Rule
Pt. 6: The Quotient Rule
Pt. 7: Parametric Differentiation
Pt. 8: Implicit Differentiation
Pt. 9: Rates of Change
AS Level
Pt. 1: First Principles
Pt. 2: Differentiation
Pt. 9: Rates of Change
Pt. 1: First Principles
Pt. 2: Differentiation
Mark Scheme
1.
5(x + 3)4 M1
2. ๐๐ฆ
๐๐ฅ=
4
1+2๐ฅ ร 2 =
8
1+2๐ฅ = 8(1 + 2x)-1 M1
๐2๐ฆ
๐๐ฅ2 = -8(1 + 2x)-2 x 2
๐2๐ฆ
๐๐ฅ2 = โ16
(1+2๐ฅ)2 M1
3.
f(x) = 4(3x + 15)0.5 M1
fโ(x) = 2(3x + 15)-0.5 x 3
fโ(x) = 6(3x + 15)-0.5 M1
4. ๐๐ฆ
๐๐ฅ = 8 โ 2e2x M1
8 โ 2e2x = 0
2e2x = 8
e2x = 4
M1
2x = ln 4
x = 1
2ln 4 = ln 2
M1
When x = ln 2
y = 8ln 2 โ 4 M1
5.
When x = 0
y = 2 + ln (1 + 0) = 2 M1
๐๐ฆ
๐๐ฅ=
1
1+4๐ฅ ร 4 =
4
1+4๐ฅ M1
When x = 0, ๐๐ฆ
๐๐ฅ = 4 M1
y = 4x + 2 M1
6.
When x = 1,
y = 1
2โln 1 =
1
2
M1
๐๐ฆ
๐๐ฅ = -(2 โ ln x)-2 ร -(
1
๐ฅ) =
1
๐ฅ(2โ๐๐๐ฅ)2 M1
When x = 1, ๐๐ฆ
๐๐ฅ =
1
1(2โln 1)2 = -4 M1
y โ 1
2= โ4(๐ฅ โ 1) M1
y = 9
2โ 4๐ฅ M1
7. ๐๐ฆ
๐๐ฅ =
2
๐ฅโ ๐ฅ2 ร (1 โ 2๐ฅ) = 2(1โ2๐ฅ)
๐ฅโ ๐ฅ2 M1
At stationary point ๐๐ฆ
๐๐ฅ = 0
2(1โ2๐ฅ)
๐ฅโ ๐ฅ2 = 0
2(1 โ 2x) = 0
M1
x = 1
2 M1
When x = 1
2
y = 2 ln(1
2 โ
1
22).
y = -4 ln 2
M1
Stationary point: (1
2, -4 ln 2) M1
8a.
โ8 โ ๐2๐ฅ = 2
8 โ e2x = 4 M1
x = 1
2ln 4 = ln 2 M1
8b. ๐๐ฆ
๐๐ฅ=
1
2(8 โ e2x)-0.5 x (-2e2x) M1
๐๐ฆ
๐๐ฅ =
โ๐2๐ฅ
โ8โ ๐2๐ฅ M1
When x = ln 2 ๐๐ฆ
๐๐ฅ =
โ๐2ln 2
โ8โ ๐2ln 2 = -2
M1
y โ 2 = -2(x โ ln 2)
2x + y = 2 + 2 ln 2 M1
2x + y = 2 + ln 22
2x + y = 2 + ln 4 M1
9a.
t = 0, N = 20
Therefore a = 20 M1
t = 8, N = 60
60 = 20e8k M1
k = 1
8ln 3 = 0.137 M1
9b.
N = 20e0.1373t M1
t = 12, N = 104 M1
9c. ๐๐
๐๐ก= 20 ร 0.1373e0.1373t = 2.747e0.1373t M1
t = 12, ๐๐
๐๐ก = 14.3 M1
N is increasing at 14.3 per second (3 s.f) M1
10a. ๐๐ฆ
๐๐ฅ = 3(5 โ 2x2)2 ร (โ4๐ฅ) M1
๐๐ฆ
๐๐ฅ = -12x(5 โ 2x2)2 M1
10b.
At stationary point ๐๐ฆ
๐๐ฅ = 0
-12x(5 โ 2x2)2 = 0 M1
x = 0
f(0) = (5 โ 2(0)2)3 = (125) M1
x = ยฑ1
2โ10
f(+1
2โ10) = 0
M1
f(-1
2โ10) = 0 M1
(-1
2โ10, 0)
(0, 125)
(1
2โ10, 0)
10c.
x = 3
2
f(3
2) = (5 โ 2(
3
2)2)3 =
1
8
M1
Gradient = -18 x 1
4 = -
9
2 M1
8y โ 1 = -36x + 54
36x + 8y โ 55 = 0 M1
11a.
fโ(x) = 3
๐ฅ - 2 M1
11b.
Gradient of curve = 4 M1 3
๐ฅ โ 2 = 4
3
๐ฅ = 6
x = 1
2
M1
11c.
Stationary point, fโ(x) = 0 M1 3
๐ฅ โ 2 = 0
x = 3
2
M1
f(3
2) = 3 ln 5(
3
2) โ 2(
3
2)
f(3
2) = 3 ln
15
2โ 3
M1
(3
2, 3 ln
15
2โ 3)
11d.
x โฅ 3
2 M1
12a.
fโ(x) = 2x โ 7 + 4
๐ฅ = 0 M1
2x2 โ 7x + 4 = 0
x = 7ยฑ โ17
4
M1
x = 0.72
x = 2.78 M1
12b.
x = 2
Therefore, y = -10
gradient = -1
M1
y + 10 = -(x-2)
y = -x - 8 M1
Differentiation
Pt. 5: Product Rule
1. Differentiate xex with respect to x. (3)
2. Find ๐๐ฆ
๐๐ฅ when y = x2 ln (x + 6) (3)
3. Find the coordinates of any stationary points of the curve y = xโ๐ฅ + 12 (4)
4. Find an equation for the tangent to the curve y = (4x โ 1)ln2x (5)
5. Find an equation for the normal to the curve y = (x2 โ 1)e3x (5)
6. The diagrams shows part of the curve with equation y = x๐๐ฅ2 and the tangent to the curve at the point P with x-
coordinate 1.
a. Find an equation for the tangent to the curve at P. (5)
b. Show that the area of the triangle bounded by this tangent and the coordinate axes is 2
3๐. (3)
7. A curve has the equation y = eโx sin x.
a. Find ๐๐ฆ
๐๐ฅ and
๐2๐ฆ
๐๐ฅ2 (4)
b. Find the exact coordinates of the stationary points of the curve in the interval -๐ โค ๐ฅ โค ๐ and determine their
nature. (8)
8. The diagram shows the curve y = f(x) in the interval 0 โค ๐ฅ โค 2๐, where f(x) = cos x sin 2x
a. Show that fโ(x) = 2 cos x(1 โ 3sin2x) (2)
b. Find the x-coordinates of the stationary points of the curve in the interval 0 โค ๐ฅ โค 2๐ (4)
c. Show that the maximum value of f(x) in the interval 0 โค ๐ฅ โค 2๐ is 4
9โ3 (4)
d. Explain why this is the maximum value of f(x) for all real values of x. (2)