Differentiation

Question 1

f(x) = x^7

Answer 1

f ‘(x) = 7x^6

Question 2

y = 1/3x^1/2

Answer 2

dy/dx = 1/6x^-1/2

Question 3

f(x) = 3/x^2 + square root x

Answer 3

f(x) = 3x^-2 + x^1/2
f ‘(x) = -6x^-3 + 1/2x^-1/2
= -6/x^3 + 1/2x6^1/2
= -6/x^3 + 1/2 square root x

Question 4

Finding the Equation of a Tangent
example
x^2 - 7x + 10 at x = 4

Answer 4

gradient = dy/dx or f ‘(x)
= 2x - 7
then sub in x = 4
= 1 (m)

point (4, y ) = sub x = 4 into
y = x^2 - 7x + 10
= (4)^2 - 7(4) + 10
= -2

equation = y -b = m(x-a)
y -(-2) = 1(x-4)
y + 2 = x - 4
y = x - 6

Question 5

Stationary Points and their nature
example
y = 2x^3 - 9x^2 + 12x

Answer 5

Differentiate
dy/dx = 6x^2 - 18x + 12

Equal to 0
Sp’s occur when dy/dx = 0
6x^2 -18x + 12 = 0

Factorise (to solve for x)
6(x^2 - 3x + 2) = 0
6(x - 2)(x - 1) = 0
x - 2 = 0 or x - 1 = 0
x = 2 or x = 1

Nature Table
sub into the differentiated equation

Comment on nature and find co-ords - into original equation
min t.p @ (1, y) max t.p @ (2, y)

sub x = 1 into the orginal equation
y = 2(1)^3 - 9(1)^2 + 12(1)
= 2 - 9 + 12
= 5

then do the same for the max t.p

Question 6

Sketch a graph
example
f(x) = x^3 + 3x^2

Answer 6

the first five steps are the same as the stationary points and their nature

roots = where it crosses the x-axis, when y = 0
f(x) = x^3 + 3x^2 = 0
= x^2(x + 3) = 0
x^2 = 0 or x = -3

y-intercept = when x = 0
f(x) = x^3 + 3x^2
= (0)^3 + 3(0)^2
= 0 (0,0)

Check for what happens at the end
sub in the large positive
(+)^3 + 3(+)^2
+ + + tends to +
sub in large negative
(-)^3 + 3(-)^2
- + + tends to -

Question 7

Optimisation
example
A(x) = 2x + 18/x

Answer 7

Differentiate
SP’s occur when x = 0
factorise
nature tables
sub into the x of the original equation

Question 8

Closed Intervals
example
f(x) = x^3 - 6x^2 + 5
-1<x<2

Answer 8

Differentiate
SP’s occure when x = 0
factorise
nature table
sub into the x of the original formula
check the edges of the domain
when x = -1 when x = 2

Question 9

Graph of the derived function
example
(1,3) (4,-2)

(1,0) (0,5) (3,0) (7,12)

Answer 9

(1,3) (4,-2) becomes (1,0) (4,0)

(1,0) (0,5) (3,0) (7,12) becomes (0,0) (3,0) (7,0)

Question 10

Chain rule

Answer 10

f(x) = (ax + b)^n
f’(x) = n(ax + b)^n - 1 x differentiated middle

Question 11

Chain rule
example
(4x^2 - 3)^-8

Answer 11

f’(x) = -8(4x^2 - 3)^-9 x 8x
= -64x(4x^2 - 3)^-9

Question 12

Differentiating sin and cos

Answer 12

sin becomes cos
cos becomes -sin
-sin becomes -cos
-cos becomes sin

Question 13

Differentiating sin and cos
y = asinbx

Answer 13

dy/dx = bacosbx

Question 14

Differentiating sin and cos
y = acosbx

Answer 14

dy/dx = -absinbx

Question 15

Differentiating sin and cos
example
f(x) = 5sin2x

Answer 15

f’(x) = 10sin2x