Chapter 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

What 2 ways are there to write the gradient function or derivative of a curve?

Answer 2

f’(x)

dy

dx

Question 3

What is the gradient function of a curve/how do you differentiate, from first principles?

Answer 3

f’(x) = lim f(x + h) - f(x)

as h →0 h

Question 4

How do you differentiate from first principles? eg. f(x) = x²

Answer 4

State formula for differentiation from first principles

f’(x) = lim f(x + h) - f(x)

as h →0 h

Sub in your f(x) eg x²

f’(x) = lim (x + h)² - x²

as h →0 h

Expand out numerator (eg. x² + 2xh + h² - x²)

All terms which do not have “h” in should cancel out

Factorise out h (ie. h (2x + h²))

Cancel out h in numerator and denominator to be left with one term which doesn’t have “h” in (ie. 2x).

State as h → 0, f’(x) = “term without h in” (ie. 2x)

At every step you must write out

f’(x) = lim numerator

as h →0 h

Question 5

If f(x) = xⁿ or y = xⁿ then what does f’(x) or dy/dx equal?

Answer 5

nx^{n - 1}

The power comes to the front and the power of x is reduced by 1.

Question 6

If f(x) = axⁿ or y = axⁿ then what is f’(x) or dy/dx?

Answer 6

anx^{n - 1}

Exactly the same, power comes to the front to be multiplied by a, and the power of x is reduced by 1.

Question 7

What must you do before you differentatiate?

Answer 7

• Ensure your function is in the format axⁿ +/- bx^m..
• Expand brackets and/or simplify to ensure you have this format
• a,n etc can be any number including negatives and fractions

Question 8

For polynominals such as f(x) = ax² + bx + c how do you differentiate?

Answer 8

Differentiate each term in turn:

f’(x) = 2ax + b

Question 9

If y = f(x) +/- g(x) what is he derivative?

Answer 9

Differentiate each term in turn

f’(x) +/- g’(x)

Question 10

How do you find the gradient of the curve at a specific point P (x₁ , y₁)?

Answer 10

• Differentiate the curve to give you the gradient function f’(x) or dy/dx
• Substitute x = x₁ into f’(x) (or dy/dx) to get the gradient at that point

Question 11

How do you find the normal to a curve at point P (x₁, y₁)?

Answer 11

• Differentiate to find the gradient function f’(x)
• Substitute in x=x₁ to find the gradient of the tangent at P
• Find the gradient of normal which is the negative reciprical of gradient of tangent ie. -1/m
• Substitute in the gradient and co-ordinates of P to find the equation of the line for the normal at P.

Question 12

How do you know if the function f(x) is increasing in the interval [a, b]?

Answer 12

Function f(x) is increasing if f’(x) is >= 0 for all values of x between a and b

Question 13

How do you know if the function f(x) is decreasing in the interval [a, b]?

Answer 13

Function f(x) is decreasing if f’(x) is <= 0 for all values of x between a and b

Question 14

What does differentiating a function f(x) twice give you?

Answer 14

f’‘(x) or d²y/dx²

Gives you the rate of change of the gradient function.

Question 15

How do you find a stationary point?

Answer 15

A stationary point is where the gradient is zero

• differentiate to find dy/dx
• set dy/dx = 0
• solve to find x
• sub x into original function to find y

Question 16

What are the three types of stationary points?

Answer 16

Minimum, maximum and point of inflection

Question 17

If f(x) has a stationary point at x = a, how do you work out whether it is a minimum or maximum?

Answer 17

• Differentiate again to get f’‘(x)
• Sub in x = a to f’‘(x)
• If f’‘(a) > 0, it is a minimum
• If f’‘(a) < 0, it is a maximum
• If f’‘(a) = 0 it could be a min, max or point of inflection. Need to investigate f’(x) either side of f’(a).

Question 18

How do you sketch the gradient function, f’(x)?

Answer 18

• At a minimum or maximum, gradient function cuts x axis
• At a point of inflection, gradient function touches x axis
• Positive gradient, gradient function is above x axis
• Negative gradient, gradient function below x axis
• Vertical asymptote, gradient function vertical asymptote
• Horizontal asymptote, gradient function horizontal asymptote at x axis.

Question 19

If V = f(t) what does dV/dt represent?

Answer 19

Rate of change of V with respect to t.

Differentiation gives you the rate of change.