Lessons 4 · 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
4
Rules for computing derivatives
Textbook, Part 1: pp. 24–29
GoalLearn to use the differentiation rules (sum, difference, constant factor, product, quotient) together with the table of derivatives.
Table: (c)′ = 0, (kx + b)′ = k, (xᵖ)′ = p·xᵖ⁻¹, (sin x)′ = cos x, (cos x)′ = −sin x, (tan x)′ = 1/cos²x, (cot x)′ = −1/sin²x, (aˣ)′ = aˣ ln a, (eˣ)′ = eˣ, (ln x)′ = 1/x, (logₐ x)′ = 1/(x ln a). Rules (when f and g have derivatives): (f ± g)′ = f′ ± g′; (c·f)′ = c·f′; (f·g)′ = f′g + fg′; (f/g)′ = (f′g − fg′)/g², where g ≠ 0. Remember that the derivative of a product is not the product of the derivatives: for f = g = x, (x²)′ = 2x while f′g′ = 1. Converting roots and fractions to powers helps: √x = x^(1/2), 1/x³ = x⁻³. For a compound expression look first at the last operation performed (sum, product, quotient) and pick the matching rule.
y = (x² + 1) sin x: by the product rule y′ = 2x sin x + (x² + 1) cos x. y = x/(x² + 1): by the quotient rule y′ = ((x² + 1) − x · 2x)/(x² + 1)² = (1 − x²)/(x² + 1)². y = x² ln x: y′ = 2x ln x + x² · 1/x = 2x ln x + x.
Class activity
“Find the rule”: the teacher writes 8 functions on cards (x³ − 4x, x² cos x, (x + 2)/x, eˣ ln x, and so on). Each group first names the rule needed for each card (“sum”, “product”, “quotient”) and then finds the derivative. Another group checks the answers.
Practice
1
Compute f′(2) for f(x) = 7x³ − 4x + 9.
80
2
Find the derivative of y = x² eˣ.
y′ = 2x eˣ + x² eˣ = eˣ(x² + 2x).
3
Find the derivative of y = (2x + 1)/(x − 3) at x = 4.
-7
4
Why is it wrong to say (x · x)′ = x′ · x′? Explain with the example.