Products and quotients, differentiated

Mathematics · Calculus & Analysis · ages 18-19

Name ______________________ Date ____________

Let h(x) = x sin x. Find h'(pi/2).
Answer: ______________
Let h(x) = (x + 1)(x + 2). Find h prime of 1.
Answer: ______________
True or false: if f(x) = x and g(x) = x, then the derivative of the product f times g equals f prime times g prime.
Circle one: True False
What is the derivative of x squared times sin x?
- 2x sin x + x squared cos x
- 2x cos x
- 2x sin x - x squared cos x
Let h(x) = (2x + 3) / (x - 1). Find h'(2).
Answer: ______________
Which of these is the derivative of f(x) = x cos x?
- cos x - x sin x
- cos x + x sin x
- -sin x
- cos x - sin x
Let h(x) = (2x + 3) / (x - 1). Find h prime of 2.
Answer: ______________
Differentiate h(x) = sin x / x squared. Which expression is h prime of x?
- (x squared cos x - 2x sin x) / x to the fourth
- (2x sin x - x squared cos x) / x to the fourth
- (cos x) / 2x
- (x squared cos x + 2x sin x) / x to the fourth
With f(2) = 3, f'(2) = 1, g(2) = 4 and g'(2) = 5, find (f times g)'(2).
Answer: ______________
What is the derivative of x cos x?
- cos x + x sin x
- -sin x
- cos x - x sin x

Answer key

For grown-ups. Fold this page away before handing over the rest.

1 · h' = sin x + x cos x, and at pi/2 this is 1 + 0 = 1.
5 · The product rule gives h prime of x = 2x + 3, and plugging in 1 gives 5. You could also multiply out first, but the product rule is faster and works even when multiplying out would be messy.
False · Multiplying first gives x squared, whose derivative is 2x, but f prime times g prime is just 1. Since 2x is not 1, the claim fails. This is exactly why the product rule exists.
2x sin x + x squared cos x · Name u = x squared and v = sin x, then add u'v and uv'.
-5 · The quotient gives -5 / (x - 1) squared, and at x = 2 the denominator is 1.
cos x - x sin x · The product rule gives two terms: 1 times cos x, plus x times the derivative of cos x, which is -sin x. Choice one matches, and it shows the minus sign that comes from differentiating cosine.
-5 · The quotient rule gives h prime of x = -5 / (x - 1) squared, and at x = 2 the denominator is 1, so the answer is -5.
(x squared cos x - 2x sin x) / x to the fourth · With the quotient rule, the numerator is (derivative of top) times bottom minus top times (derivative of bottom), all over the bottom squared. Only choice one has both the correct order and the correct denominator x to the fourth.
19 · No formulas needed: 1 times 4 + 3 times 5 = 19.
cos x - x sin x · The product rule gives 1 times cos x plus x times -sin x.