3. Techniques of Differentiation

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

y = tan x + cot x

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

y = sin x / 1 + cos 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.

At what times and positions is the velocity zero?

Find the derivative of the following functions.

y = sin x + cos x

A weight is attached to a spring and reaches its equilibrium position (x = 0). It is then set in motion resulting in a displacement of x = 10 cos t, where x is measured in centimeters and t is measured in seconds. See the accompanying figure.

b. Find the spring’s velocity when t = 0, t = π/3, and t = 3π/4.

Find the derivative of the function.

$y=\frac{\cot \theta }{3+\sec \theta }$

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

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

Let f(x) = sin x. What is the value of f′(π)?

Is there a value of b that will make

g(x) = { x + b, x < 0

cos x, x ≥ 0

continuous at x = 0? Differentiable at x = 0? Give reasons for your answers.

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

y = cos² x

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.

Find the derivative of the function.

$f\left(x\right)=\frac{5\cos x}{2x^3}$