Differentiation

Question 0

Implicit differentiation: Chain rule:

Answer 0

d dy
— y(^n) = ny(^n-1) —
dx dx

                         dy e.g. y(^3) = 3y(^2)---- 
                         dx

Question 1

Finding gradient of a curve with parametric coordinates:

Answer 1

• Differentiate x and y with respect to t
  dy dy dx
• Rearrange into — = — / —
  dx dt dt
• Substitude given value back into equation

Question 2

Implicit differentiation: Product rule:

Answer 2

d dy
— (xy) = x —- + y
dx dx

                             dy e.g. 4xy(^2) = 4x*2y ---- + 4y(^2) 
                             dx

Question 3

In an implicit equation, when f(y) is differentiated with respect to x it becomes:

Answer 3

dy
f’(y) —
dx

Question 4

In an implicit equation, a product term such as f(x)*g(x) is differentiated by the product rule becomes:

Answer 4

dy
f(x) * g’(y) —- + g(y) * f’(x)
dx

Question 5

What are two examples of an implicit relation?

Answer 5

x^2 + y^2 = 8x

cos(x + y) = siny

Question 6

How do you connect rates of change in a question involving more than two variables?

Answer 6

Use the chain rule once or several times:
dA dA dr
e.g. —- = —- * —-
dt dr dt

Question 7

Implicit differentiation refers to the left side of the equation if it gas got a power or a product

Answer 7

You must then rearragne to make dy/dx the subject of the formula.