3. Techniques of Differentiation

23–51. Calculating derivatives Find the derivative of the following functions.

y = cos² x

23–51. Calculating derivatives Find the derivative of the following functions.

y = sin x / 1 + cos x

23–51. Calculating derivatives Find the derivative of the following functions.

y = tan x + cot x

Find the derivative of the following functions.

y = cot x / (1 + csc x)

An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.

Find the velocity of the oscillator, v(t) =y′(t).

An object oscillates along a vertical line, and its position in centimeters is given by y(t)=30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.

At what times and positions is the velocity zero?

An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sin t - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.

The acceleration of the oscillator is a(t) = v′(t). Find and graph the acceleration function.

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

d/dx (sec x) = sec x tan x

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