Lessons 3 · 1 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 11, Parts 1 and 2, 1st edition. ZAMIN NASHR, Tashkent, 2018
3
The derivative and its geometric and physical meaning
Textbook, Part 1: pp. 16–23
GoalState the definition of the derivative as a limit, understand its geometric meaning (slope of the tangent) and physical meaning (instantaneous velocity), and find simple derivatives from the definition.
New words
derivative · hosiladifferentiation · differensiallashinstantaneous velocity · oniy tezlikslope of the tangent · urinmaning burchak koeffitsiyenti
Explanation
The derivative of y = f(x) at x is the limit f′(x) = lim (f(x + h) − f(x))/h as h → 0, if it exists; finding the derivative is called differentiation. Notation: f′(x), y′ or dy/dx. Geometric meaning: f′(x₀) equals the slope of the tangent to the graph at (x₀, f(x₀)). Physical meaning: if a point moves by s(t), its instantaneous velocity at time t is v(t) = s′(t); in general the derivative is the rate of change of the function. From the definition we get (c)′ = 0, (x)′ = 1, (x²)′ = 2x, (x³)′ = 3x². If the derivative is positive the tangent rises (the function increases near that point); if negative the tangent falls.
Worked examples
f(x) = 3x² − 2x. By definition: f(x + h) − f(x) = 3(2xh + h²) − 2h = 6xh + 3h² − 2h. Dividing by h: 6x + 3h − 2. As h → 0, f′(x) = 6x − 2. For instance f′(1) = 4: the tangent at (1, 1) has slope 4.
f(x) = 1/(x + 1), x ≠ −1. f(x + h) − f(x) = 1/(x + h + 1) − 1/(x + 1) = −h/((x + h + 1)(x + 1)). Dividing by h and letting h → 0 gives f′(x) = −1/(x + 1)². The derivative is negative everywhere, so the function decreases on each interval of its domain.
Class activity
“Derivative twins”: split the class in two. One group picks a function (say x³ or 1/(x + 1)) and finds its derivative from the definition; the other group estimates the tangent slope at one point from a table (h = 0.1, 0.01, 0.001). The groups then compare results.
Practice
1
Using the definition, find the derivative of f(x) = 5x − 2.
(5(x + h) − 2 − 5x + 2)/h = 5, so f′(x) = 5.
2
A point moves by s(t) = 4t² − t (m). Given s(t + h) − s(t) = 8th + 4h² − h, find the instantaneous velocity (m/s) at t = 2 s.
15
3
Find the slope of the tangent to y = x² − 3x at x = 2.
1
4
If f′(x₀) < 0, what can be said about the graph and the tangent? Why?
The tangent’s slope is negative, so it falls from left to right; the function decreases near x₀, because the difference quotient (f(x₀ + h) − f(x₀))/h is negative.