M - Pure 1 - 12) Differentiation

Question 1

Equation for differentiation by first principles

Answer 1

f’(x) = lim_h→0 [f(x + h) - f(x)]/h

Question 2

The derative of every power of x and constants

Answer 2

• ax^n≥2 → anx^n-1
• bx → x
• c → 0

Question 3

Effect on the gradient of a normal

Answer 3

Negative reciprocal

Question 4

Thing that shows when a function is increasing or decreasing on the interval (a, b)

Answer 4

• When increasing → f’(x) ≥ 0 for all values of x
• When decreasing → f’(x) ≤ 0 for all values of x

Question 5

Notations of second order derivatives

Answer 5

• f’‘(x)
• d²y/dx²

Question 6

Meaning of stationary point

Answer 6

A point where f’(x) = 0

Question 7

Thing that shows what type of stationary point x = a is

Answer 7

• Local minimum → if f’‘(a) > 0
• Local maximum → if f’‘(a) < 0
• If f’‘(a) = 0 → the point could be a local minimum, local maximum or point of inflection - look at points on either side

Question 8

Features of a function and the corresponding sketches of its gradient function

Answer 8

• Positive gradient → above x-axis
• Negative gradient → below x-axis
• Maximum or minimum → cuts x-axis
• Point of inflection → touches x-axis
• Vertical asymptote → vertical asymptote
• Horizontal asymptote → horizontal asymptote at x-axis