Derivatives

Question 1

Derivative from 1st Principles

Answer 1

= lim as h->0, f(x+h) - f(x) / h

Question 2

Define differentiable

Answer 2

At a point, the function must be continuous and the slopes of the secants from both sides must be equal.

Question 3

The Constant Rule

Answer 3

If f(x)=c then f1=0

Question 4

The Constant Multiplier Rule

Answer 4

If f(x)=cg(x) then f1=cg1

Question 5

The Sum/Difference Rule

Answer 5

If f(x)=g(x)+-h(x) then f1=g1+-h1

Question 6

The Product Rule

Answer 6

If y=f(x)g(x) and both are differentiable, then

y[1]=f1g(x) + g1f(x)

Question 7

How to prove Product Rule?

Answer 7

Insert -f(x)g(x+h)+f(x)(g(x+h) into middle of numerator, then factor by grouping.

Question 8

Relationship of D, V, and a

Answer 8

D = s(t), V = s1, a = V1 or s2

Question 9

The Chain Rule

Answer 9

If f(x)=g(h(x)) then f1=g1*h1

Question 10

The Quotient Rule

Answer 10

If Q(x) = f(x)/g(x) then Q1= f1g(x) - g1f(x) / g(x)^2

Question 11

How to prove the Quotient Rule?

Answer 11

Insert -f(x)/g(x+h) after first part and +f(x)/g(x+h) after second, common factor both halves, multiply all 1s by g(x), change to f1 and g1.

Question 12

Demand / Price Function

Answer 12

p(x): price for x units sold.

Question 13

Revenue Function

Answer 13

R(x)= (# units sold) (price)
R(x)= xp(x)

Question 14

Cost Function

Answer 14

usually given. cost to produce/buy x units

Question 15

Profit Function

Answer 15

P(x)= R(x)-C(x)
P(x)= xp(x)-C(x)