Name That Rule
What differentiation rule is required to differentiate
y = (3x + 2)5?
Chain Rule
Find dy/dx if
y = 5x4 − 3x2 + 7x − 1.
dy/dx = 20x3 − 6x + 7
A student differentiates
y = (3x + 2)4 and obtains
y' = 4(3x + 2)3.
What did the student forget?
They forgot the derivative of the inside function.
Correct answer:
y' = 12(3x + 2)3
Without differentiating, identify the rule required for y = (x2 + 5)7.
Chain Rule
Find dy/dx if y = x3(x2 + 1)2.
dy/dx = x2(x2 + 1)(7x2 + 3)
What differentiation rule is required to differentiate
y = x3(x2 + 1)?
Product Rule
Find dy/dx if
y = (2x − 3)5.
dy/dx = 10(2x − 3)4
A student differentiates
y = x2(x + 1) and obtains
y' = 2x(x + 1). What is the mistake?
They differentiated only the first factor.
The Product Rule is required:
y' = 2x(x + 1) + x2
Without differentiating, identify all the rules required for
y = x3(x − 4)2.
Product Rule and Chain Rule
Differentiate:
y = (x2 - 1)3/x
y' = ((x2-1)2(5x2 + 1))/x2
What differentiation rule is required to differentiate
y = (x2 + 1)/(x − 3)?
Quotient Rule
Find dy/dx if
y = x2(x + 1)3.
dy/dx = x(x + 1)2(5x + 2)
A student differentiates
y = (x2 + 1)/(x − 1) and obtains
y' = 2x. What rule has been incorrectly applied?
They did not use the Quotient Rule.
For y = (x2 + 1)3/(x − 2), which differentiation rule should you consider first?
Quotient Rule.
You will then need the Chain Rule when differentiating the numerator.
Find dy/dx if
y = (2x + 1)3(x2 − 4)2.
dy/dx = 6(2x + 1)2(x2 − 4)2 + 4x(2x + 1)3(x2 − 4)
Identify all the differentiation rules required to differentiate
y = (x2 + 1)4(x − 2)3.
Product Rule and Chain Rule
Find dy/dx if
y = (x2 + 1)3/x.
dy/dx = (x2 + 1)2(5x2 − 1)/x2
A student differentiates
y = (x2 + 1)3 and obtains
y' = 3(x2 + 1)2. Identify the error and give the correct derivative.
They forgot the derivative of the inner function, x2 + 1.
Correct:
y' = 6x(x2 + 1)2
A student says: "For y = (x2 + 1)3(x − 4)5, I only need the Chain Rule." Is the student correct? Explain.
No.
The Product Rule is required because the function is a product. The Chain Rule is also required for both factors.
Find dy/dx if
y = x2√(x + 1)/(x − 2)2.
dy/dx = x(3x2 − 5x − 4) / [2(x − 2)3√(x + 1)]
Identify all the differentiation rules required to differentiate
y = x2(3x − 1)4/(x + 2)3.
Product Rule, Quotient Rule and Chain Rule
Find dy/dx if
y = (2x − 1)4/(x2 + 1)3.
dy/dx = 2(2x − 1)3(−x2 + 8x + 3)/(x2 + 1)4
A student differentiates y = x2(x2 + 1)4 and obtains
y' = 2x(x2 + 1)4 + 4x2(x2 + 1)3.
Identify the error and give the correct derivative.
The Product Rule was used correctly, but the Chain Rule was forgotten in the second term.
Correct:
y' = 2x(x2 + 1)4 + 8x3(x2 + 1)3
Before differentiating y = x2(3x − 1)4/(x + 2)3, state all the rules you will need and describe your overall strategy.
Product Rule, Quotient Rule and Chain Rule.
Use the Quotient Rule for the numerator and denominator. Within the numerator, use the Product Rule. Use the Chain Rule for the composite functions.
Find dy/dx and state all the differentiation rules used:
y = (x2 + 1)3(2x − 1)4 / [x2(x + 3)3]
Rules used: Product Rule, Quotient Rule and Chain Rule.
dy/dx =
2(x2 + 1)2(2x − 1)3
(8x5 + 18x4 − 20x3 − 51x2 + 6x + 3)
/
[x3(x + 3)4]