3. Techniques of Differentiation

A differential equation is an equation involving an unknown function and its derivatives. Consider the differential equation y′′(t)+y(t) = 0.

b. Show that y = B cos t satisfies the equation for any constant B.

Find the first and second derivatives of the functions in Exercises 33–38.

w = ((1 + 3z) / 3z) (3 − z)

Find y'' if:

a. y = csc x

Derivative Calculations

In Exercises 1–12, find the first and second derivatives.

y = 6x² − 10x − 5x⁻²

Find y'' for the following functions.

y = tan x

By computing the first few derivatives and looking for a pattern, find the following derivatives.

c. d⁷³/dx⁷³ (x sin x)

Find y'' for the following functions.

y = x sin x

Explain why or why not Determine whether the following statements are true and give an explanation or counter example.

b. d²/dx² (sin x) = sin x.

Derivative Calculations

In Exercises 1–12, find the first and second derivatives.

r = 12/θ − 4/θ³ + 1/θ⁴

By computing the first few derivatives and looking for a pattern, find the following derivatives.

b. d¹¹⁰/dx¹¹⁰ (sin x − 3 cos x)

Find the derivatives of all orders of the functions in Exercises 29–32.

y = x⁵ / 120

Find the first and second derivatives of the functions in Exercises 33–38.

s = (t² + 5t − 1) / t²

Find y⁽⁴⁾ = d⁴y/dx⁴ if:

a. y = −2 sin x

If f(t)=t¹⁰, find f′(t), f′′(t), and f′′′(t).

Find f′(x), f′′(x), and f′′′(x) for the following functions.

f(x) = 3x² + 5e^x