Time: 20 minutes

Calculus PracticeTest Solutions

Name: 1. (7 points) Starting with the definition of tan(x) = sin(x)/ cos(x), show that the derivative of f (x) = tan(x) is f 0 (x) = sec2 (x). Solution

d d sin(x) tan(x) = dx dx cos(x) cos(x) cos(x) − sin(x) (− sin(x)) = cos2 (x) 1 = cos2 (x) = sec2 (x)

2. (8 points) Use implicit differentiation to compute y 0 for 2 sin(x) tan(y) = 0. Solution Differentiating both sides with respect to x yields 2 sin(x) · sec2 (y)y 0 + tan(y) · 2 cos(x) = 0. Then we solve for y 0 as follows:

2 sin(x) · sec2 (y) · y 0 + tan(y) · 2 cos(x) = 0 sin(x) · sec2 (y) · y 0 = − tan(y) cos(x) y0 =

− tan(y) · cos(x) sin(x) · sec2 (y)

√

3. (5 points) Compute the derivative of g(x) =

x2 +1 4x−2 .

Solution We must use the quotient rule. We have −1

(4x − 2)( 12 (x2 + 1) 2 · (2x)) − g (x) = (4x − 2)2 0

√

x2 + 1 · 4

.

1

If you wanted to simplify this, a good first step would be to multiply by

http://math.furman.edu/PracticeTest/

(x2 +1) 2 1 (x2 +1) 2

.

c

2002 Mark R. Woodard