Math 121: Derivative Worksheet

Math 121: Derivative Worksheet
PART 1: DERIVATIVE FORMULAS. Complete the given statements
• For any constant c,
•
d
(c · f (x)) =
dx
d
(f (x) + g(x)) =
dx
d
(f (x) · g(x)) =
dx
d f (x)
•
=
dx g(x)
•
•
d
[f (g(x))] =
dx
• For any constant c,
d
(c) =
dx
• For any real number n,
d n
(x ) =
dx
• Suppose b > 0 and b 6= 1, then
•
d x
(b ) =
dx
d x
(e ) =
dx
• Suppose b > 0 and b 6= 1, then
•
d
(ln x) =
dx
•
d
(sin x) =
dx
•
d
(cos x) =
dx
•
d
(tan x) =
dx
d
(logb x) =
dx
1
•
d
(cot x) =
dx
•
d
(sec x) =
dx
•
d
(csc x) =
dx
•
d
sin−1 x =
dx
•
d
cos−1 x =
dx
•
d
tan−1 x =
dx
PART 2: DERIVATIONS OF DIFFERENTIATION FORMULAE
1. Suppose x is measured in radians. Use the definition of the derivative to show that
d
(cos x) = − sin x.
dx
2. Use the quotient rule to prove that
d
(cot x) = − csc2 x
dx
3. Consider y = cos−1 (x)
(a) Specify the domain & range restrictions of y = cos−1 (x).
d
−1
(b) Derive the formula
by rewriting y = cos−1 (x) as cos y =
(cos−1 (x)) = √
2
dx
1−x
x and then differentiating implicitly.
4. Suppose that f is an invertible, differentiable function.
(a) Beginning with the fact that f (f −1 (x)) = x, derive a formula for the derivative
of f −1 (x).
(b) Now suppose f (x) = cos x and f −1 (x) = cos−1 x. Use your fact from part (a) to
d
−1
justify that
(cos−1 (x) = √
.
dx
1 − x2
2
PART 3: DERIVATIVE CALCULATIONS & APPLICATIONS
5. Differentiate
(a) y = tan(e−3x )
(b) y = ln (cos (2x − 1))
(c) y = xcos x
2
(d) y = earctan 3x
r
1 − sin x
(e) y =
1 − cos x
2x + 1
(f) y = ln √
x − 2 tan2 x
(g) y = x2 csc x
6. Find the value(s) of x at which the tangent line to the graph of y = ln (x2 + 11) is
perpendicular to y = −6x + 5.
7. Find all x where the the curve y = x2 (3x + 7)5 has horizontal tangent lines.
8. Find all x where the tangent line to y = arctan x is parallel to −x + 2y = 7. Sketch
the graph of y = arctan x and the tangents at these values of x.
9. Use the given table to answer the following questions.
x f (x) f 0 (x)
−1
2
3
2
0
4
g(x)
2
1
g 0 (x)
−3
−5
(a) Let h(x) = f (g(x)). Compute h0 (−1).
(b) Let h(x) = [f (x)]2 . Compute h0 (2).
(c) Let h(x) = [g(f (x))]3 . Compute h0 (−1).
10. Consider the curve 2x2 − 3y 2 = 4. Express your answer only in terms of x and y.
dy
.
dx
d2 y
(b) Calculate 2 .
dx
(a) Calculate
π
11. Find the slope of the tangent line to the curve sin (xy) = x at the point 1,
.
2
3
12. The set of ordered pairs (x, y) which satisfy the equation (x2 + y 2 − x)2 = x2 + y 2 form
the curve shown below, called a cardioid.
Let L1 be the line which is tangent to the curve at the point (0, 1) and let L2 be the
line which is tangent to the curve at the point (0, −1). At which point in the xy-plane
do L1 and L2 intersect?
4