3. Techniques of Differentiation

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

a. d⁹⁹⁹/dx⁹⁹⁹ (cos x)

Second Derivatives

Find y'' in Exercises 59–64.

y = (1 − √x)⁻¹

Find y⁽⁴⁾ = d⁴y/dx⁴ if:

b. y = 9 cos x

Find y'' for the following functions.

y = cos θ sin θ

Let $f\left(z\right)=\sin \left(z\right)$. What is the ${43}^{rd}$ derivative of $f\left(z\right)$?

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

f(x) = 3x³ + 5x² + 6x

Find d²/dx² (sin x + cos x).

Find y'' for the following functions.

y = e^x sin x

Find the third derivative of the given function.

$f\left(x\right)=24+4x^5$

Find the third derivative of the given function.

$f(t)=5t^{-4}$

Second Derivatives

Find y'' in Exercises 59–64.

y = x(2x + 1)⁴

Derivative Calculations

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

y = x² + x + 8

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

f(x) = (x² - 7x - 8) / (x + 1)

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 = x³/3 + x²/2 + x/4