Derivative and Its Applications

Table of Derivatives and Rules

The derivative shows the rate of change of a function.

• (c)′ = 0, (x)′ = 1, (xⁿ)′ = n · xⁿ⁻¹
• (sin x)′ = cos x, (cos x)′ = −sin x
• (tg x)′ = 1 / cos²x, (√x)′ = 1 / (2√x)
• (eˣ)′ = eˣ, (aˣ)′ = aˣ · ln a, (ln x)′ = 1 / x

Rule	Formula
Sum	(u + v)′ = u′ + v′
Product	(u · v)′ = u′ · v + u · v′
Quotient	(u / v)′ = (u′ · v − u · v′) / v²
Composite function	(f(g(x)))′ = f′(g(x)) · g′(x)

Applications

• If f′(x) > 0 on an interval, the function increases; if f′(x) < 0, it decreases.
• A point where f′ changes sign from plus to minus is a local maximum; if f′ changes sign from − to +, it is a minimum.
• Tangent line: y = f(x₀) + f′(x₀) · (x − x₀). The slope equals f′(x₀).
• In motion, v(t) = s′(t), a(t) = v′(t).

1. Find f′(x)
2. Find the critical points from the equation f′(x) = 0
3. Check the sign of f′ around each point
4. If there is a closed interval, also calculate the values at the endpoints

Graph example: the parabola y = x² − 4x + 5 and its tangent line at the point x = 3, y = 2x − 4.

x	1	2	3	4	5
y = x² − 4x + 5	2	1	2	5	10
tangent 2x − 4	−2	0	2	4	6

The two graphs touch only at the point (3; 2); everywhere else the parabola lies above the tangent line.

Examples

• f(x) = x³ − 3x² + 2: f′ = 3x(x − 2). At x = 0 there is a local maximum f(0) = 2, at x = 2 a local minimum f(2) = −2. These are not the greatest and smallest values: as x → +∞, f(x) → +∞, and as x → −∞, f(x) → −∞.
• y = x² − 4x + 5, x₀ = 3: f(3) = 2, f′(3) = 2, y = 2 + 2(x − 3) = 2x − 4.
• f(x) = x³ − 12x, interval [−3; 3]: from f′ = 0 we get x = ±2. f(−3) = 9, f(−2) = 16, f(2) = −16, f(3) = −9. The greatest value is 16, the smallest is −16.

Be Careful

• (sin 2x)′ = 2cos 2x, not cos 2x.
• f′(x) = 0 is not always an extremum: for f(x) = x³, f′(0) = 0, but there is no extremum.
• On a closed interval, check the endpoints too.
• For a tangent line, the slope is f′(x₀), not f(x₀).