Basic Differentiation
Slightly more advanced rules
Properties of Curves
100

The derivative calculates the ________.

Slope or rate of change

100

Suppose y = —2x2(x + 4). For what values of x does dy/dx = 10?

x = -1 or -5/3

100

State the relationship between speed and velocity.

Speed is the absolute value of velocity, i.e. velocity without direction.

200
d/dx 3x²-x+3 =
6x-1
200

Find the value of z for which the tangent to f(x) = ax⋅ √(1 - x) has gradient a.

x = 0

See HL2 17C #7b

200

Find and classify the stationary points of f(x) = x2 / (x + 3).

Local maximum (-6, -12)

Local minimum (0, 0)

Note: x = -3 is a critical point (f' undefined), but not a stationary (f' = 0) nor inflection point (f'' = 0 and change signs).

300
Differentiate y=√13x²-5x+8
y′ =26x-5/ 2√13x²-5x+8
300

y = x3, show that (dy/dx)⋅(dx/dy) = 1

See HL2 17B.2 #8

300

Find intervals where f(x) = -x3 - 6x2 +36x -17 is increasing.

-6 ≤ x ≤ 2

400

The cost of producing x toasters each week is given by C = 1785+3x+0.002x2 pounds. Find the value of dC/dx when x = 1000, and interpret its meaning.

0.004 (1000) + 3 = 7 pounds. When 1000 toasters are being produced each week, the cost increases by another 7 pounds per toaster.

400

The gradient function of f(x) = (2x - b)a is
f'(x) = 24x2 — 24x +6. Find the constants a and b.

a = 3, b = 1

400

Find the equation of the normal to f(x) = x + lnx where x = 1.

x + 2y = 3

500

Using quotient rule, differentiate y= (1 + lnx) / (x² - lnx)

y′= [(1/x) - x - 2xlnx] / (x²-lnx)²

500

The tangent to f(x) = 2x + 4x at point P has gradient ln2. Find the coordinates of P.

(-1, 3/4)

See HL2 17E.2 #4

500

For the function f(x) = ln(x2 + 5), find the 

a. turning points

b. points of inflection

c. intervals where it's concave up or down

a. (0, ln5) minimum

b. (-√5, ln10) and (√5, ln10) are non-stationary inflection points

c. Concave up for -√5 ≤ x ≤ √5, down everywhere else.

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