3. Techniques of Differentiation

Find y⁽⁴⁾ = d⁴y/dx⁴ if:

a. y = −2 sin x

Find y'' if:

a. y = csc x

Find y⁽⁴⁾ = d⁴y/dx⁴ if:

b. y = 9 cos x

Derivative Calculations

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

y = x² + x + 8

Derivative Calculations

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

w = 3z⁷ − 7z³ + 21z²

Derivative Calculations

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

y = 6x² − 10x − 5x⁻²

Derivative Calculations

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

r = 12/θ − 4/θ³ + 1/θ⁴

Second Derivatives

Find y'' in Exercises 59–64.

y = (1 − √x)⁻¹

Second Derivatives

Find y'' in Exercises 59–64.

y = x(2x + 1)⁴

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

w = ((1 + 3z) / 3z) (3 − z)

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

a. d⁹⁹⁹/dx⁹⁹⁹ (cos x)

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

b. d¹¹⁰/dx¹¹⁰ (sin x − 3 cos x)

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

c. d⁷³/dx⁷³ (x sin x)