3. Techniques of Differentiation

Derivative Calculations

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

y = x³/3 + x²/2 + x/4

Find the third derivative of the given function.

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

Find the third derivative of the given function.

$y=3x^2+9x+1$

Find the third derivative of the given function.

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

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

f(x) = 3x³ + 5x² + 6x

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

f(x) = 3x² + 5e^x

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

f(x) = (x² - 7x - 8) / (x + 1)

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

Find d²/dx² (sin x + cos 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.

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 y'' for the following functions.

y = x sin x

Find y'' for the following functions.

y = e^x sin x

Find y'' for the following functions.

y = tan x

Find y'' for the following functions.

y = cos θ sin θ