3. Techniques of Differentiation

9–61. Evaluate and simplify y'.

y = (sin x / cos x+1)^1/3

9–61. Evaluate and simplify y'.

y = csc⁵ 3x

9–61. Evaluate and simplify y'.

y = tan (sin θ)

9–61. Evaluate and simplify y'.

y=sin √cos² x+1

Find an equation of the line tangent to the following curves at the given value of x.

y = 1+2 sin x; x = π/6

Find an equation of the line tangent to the following curves at the given value of x.

y = csc x; x = π/4

{Use of Tech} Difference quotients Suppose f is differentiable for all x and consider the function D(x) = f(x+0.01)-f(x) / 0.01 For the following functions, graph D on the given interval, and explain why the graph appears as it does. What is the relationship between the functions f and D?

f(x) = sin x on [−π,π]

A race Jean and Juan run a one-lap race on a circular track. Their angular positions on the track during the race are given by the functions θ(t) and ϕ(t), respectively, where 0≤t≤4 and t is measured in minutes (see figure). These angles are measured in radians, where θ=ϕ=0 represent the starting position and θ=ϕ=2π represent the finish position. The angular velocities of the runners are θ′(t) and ϕ′(t). <IMAGE>

b. Which runner has the greater average angular velocity?

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)

lim x→π/4 cot x−1 / x−π/4

Derivatives

In Exercises 1–18, find dy/dx.

y = (sec x + tan x)(sec x − tan x)

Derivatives

In Exercises 1–18, find dy/dx.

f(x) = x³ sin x cos x

Derivatives

In Exercises 23–26, find dr/dθ.

r = θ sin θ + cos θ

Derivatives

In Exercises 23–26, find dr/dθ.

r = (1 + sec θ) sin θ

Derivatives

In Exercises 27–32, find dp/dq.

p = (sin q + cos q) / cos q

Derivatives

In Exercises 27–32, find dp/dq.

p = (q sin q) / (q² − 1)