Lesson test: The rules of differentiation

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The rules of differentiation

1.[2p]
Differentiate $(x^{2} + 1) (x^{3} - 2 x)$ and evaluate at $x = 2$ .

CorrectNot quite: 66

2.[3p]
Differentiate $\frac{x}{x^{2} + 1}$ and evaluate at $x = 2$ .

CorrectNot quite: -0.12

3.[2p]
Differentiate $(x^{3} + 1)^{5}$ and evaluate at $x = 1$ .

CorrectNot quite: 240

4.[3p]
Differentiate $\sqrt{x^{2} + 9}$ and evaluate at $x = 4$ .

CorrectNot quite: 0.8

5.[2p]
What single test disproves the guess that $(f g)' = f' g'$ ?

Take $f = g = x$ : the product differentiates to $2 x$ , while the product of the derivatives is $1$

Correct

Take $f = g = 1$ : both sides are zero, so the rule is undefined

The answer is: Take $f = g = x$ : the product differentiates to $2 x$ , while the product of the derivatives is $1$

The guess has the wrong units

The answer is: Take $f = g = x$ : the product differentiates to $2 x$ , while the product of the derivatives is $1$

The guess fails only for trigonometric functions

The answer is: Take $f = g = x$ : the product differentiates to $2 x$ , while the product of the derivatives is $1$

6.[1p]
The power rule $\frac{d}{d x} x^{n} = n x^{n - 1}$ holds for negative integer exponents as well as positive ones.

True

Correct

False

The answer is: True

7.[3p]
Match each rule to the structure it differentiates.

Product rule
Quotient rule
Chain rule
Power rule

one function divided by another
one function inside another
a single variable raised to an exponent
two factors multiplied

Show the answer

Product rule: two factors multiplied Quotient rule: one function divided by another Chain rule: one function inside another Power rule: a single variable raised to an exponent

8.[3p]
Which of these statements are correct?

Select all that apply

The derivative of a constant is zero, so adding a constant never changes a derivative

Correct

Differentiation is linear, so a polynomial may be differentiated term by term

Correct

The chain rule contributes one factor per nested layer

Correct

The derivative of a quotient is the quotient of the derivatives

The answer is: The derivative of a constant is zero, so adding a constant never changes a derivative, Differentiation is linear, so a polynomial may be differentiated term by term, The chain rule contributes one factor per nested layer

9.[2p]
Why do the rules of this lesson fail to reach $\sin x$ and $2^{x}$ ?

Neither is built from powers by adding, multiplying, dividing or composing, so their difference quotients need new limits

Correct

Both are discontinuous somewhere

The answer is: Neither is built from powers by adding, multiplying, dividing or composing, so their difference quotients need new limits

Both are inverse functions

The answer is: Neither is built from powers by adding, multiplying, dividing or composing, so their difference quotients need new limits

Neither has a derivative

The answer is: Neither is built from powers by adding, multiplying, dividing or composing, so their difference quotients need new limits

1.[2p] Differentiate (x2+1)(x3−2x) and evaluate at x=2.

2.[3p] Differentiate xx2+1 and evaluate at x=2.

3.[2p] Differentiate (x3+1)5 and evaluate at x=1.

4.[3p] Differentiate x2+9 and evaluate at x=4.

5.[2p] What single test disproves the guess that (fg)′=f′g′?

6.[1p] The power rule ddxxn=nxn−1 holds for negative integer exponents as well as positive ones.

7.[3p] Match each rule to the structure it differentiates.

8.[3p] Which of these statements are correct?

9.[2p] Why do the rules of this lesson fail to reach sinx and 2x?

1.[2p]
Differentiate $(x^{2} + 1) (x^{3} - 2 x)$ and evaluate at $x = 2$ .

2.[3p]
Differentiate $\frac{x}{x^{2} + 1}$ and evaluate at $x = 2$ .

3.[2p]
Differentiate $(x^{3} + 1)^{5}$ and evaluate at $x = 1$ .

4.[3p]
Differentiate $\sqrt{x^{2} + 9}$ and evaluate at $x = 4$ .

5.[2p]
What single test disproves the guess that $(f g)' = f' g'$ ?

6.[1p]
The power rule $\frac{d}{d x} x^{n} = n x^{n - 1}$ holds for negative integer exponents as well as positive ones.

7.[3p]
Match each rule to the structure it differentiates.

8.[3p]
Which of these statements are correct?

9.[2p]
Why do the rules of this lesson fail to reach $\sin x$ and $2^{x}$ ?