3. Techniques of Differentiation

Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.

d/dx (csc x) = -csc x cot x

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/2 cos x/x−π/2

For what values of x does g(x) = x−sin x have a slope of 1?

Use a graphing utility to plot the curve and the tangent line.

y = cos x / 1−cos x; x = π/3

Using identities Use the identity sin 2x=2 sin x cos x sin 2 to find d/dx (sin 2x). Then use the identity cos 2x = cos² x−sin² x to express the derivative of sin 2x in terms of cos 2x.

5-8. Use differentiation to verify each equation.

d/dx (tan³ x-3 tan x+3x) = 3 tan⁴x

9–61. Evaluate and simplify y'.

y = 5t² sin t

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?