12. Differentiation

Question 1

What is the gradient of a curve at a given point defined as?

Answer 1

The gradient of the tangent to the curve at that point

Question 2

How do you determine the derivative from first principles?

Answer 2

See card

Question 3

What is the formula for differentiating?

Answer 3

See card

Question 4

What is the normal?

Answer 4

The line perpendicular to a tangent

Question 5

How do you work out whether a point is a local maximum, local minimum or a point of inflection?

Answer 5

Determine the stationary points
f’(x - h) = positive + f’(x + h) = negative: local maximum
f’(x - h) = negative + f’(x + h) = positive: local minimum
f’(x - h) + f’(x + h) = both positive / negative: inflection

Question 6

How do you work out the rate of change of a function?

Answer 6

Differentiate twice to find the second derivative
d2y
—–
dx2

Question 7

How do you model with differentiation?

Answer 7

dy / dx = small change in y / small change in x

This represents the rate of change