Rules for computing derivatives
Answers are for parents and teachers.
1
Find the derivative of f(x) = x⁴ − 6x² + 3x − 8 and compute f′(−2).
-5
2
Write the derivative of y = (x³ + 2) cos x.
y′ = 3x² cos x − (x³ + 2) sin x.
3
Find the derivative of y = ln x / x (x > 0) and show that it equals zero at x = e.
y′ = (1/x · x − ln x · 1)/x² = (1 − ln x)/x². At x = e, ln e = 1, so y′ = 0.
4
Find the derivative of y = 3eˣ − 2 cos x + 5/x.
y′ = 3eˣ + 2 sin x − 5/x².
5
At which points of [0, 2π] does the derivative of y = x − 2 sin x vanish?
y′ = 1 − 2 cos x. y′ = 0 ⇔ cos x = 1/2 ⇔ x = π/3 or x = 5π/3.