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Differentiation rules: differentiation of elementary functions d d d -{f(x) g(x)}= f(x) 8(x) + g(x). dx f(x),, assuming f and g are differentiable. dx 4.12. Prove the formula dx By definition, d f(x +Ax) g (x +Ax) – f(x) g(x) {f(x) g(x)}= lim dx Ax-0 Ax f(x+Ax) {g (x +Ax) – g(x)}+g(x) {f(x+Ax) – f(x)} lim Ax-0 Ax g(x +Ax) - g(x) f(x+ Ax) - f(x) = lim f(x +Ax) + lim g(x) Ax→0 Ax Ax→0 Ax d d = f(x) 8(x)+ g(x) f(x) dx dx Another method: Let u = f(x), v = g(x). Then Au = f(x + Ax) – f(x) and A v = g(x + Ax) – g(x); i.e., f(x + Ar) = u + Au, g(x + Ar) = v + Av. Thus, d uv = lim (и + Ди)(0 + дu) — w uAv +vAu + AuAv lim dx Ar→0 Ar Ar0 Ax dv du Δυ = lim u Ar Au Au Av = u- + Ax Ar-0 Ar dx dx where it is noted that Au →0 as Ar → 0, since v is supposed differentiable and thus continuous.

Differentiation rules: differentiation of elementary functions d d d -{f(x) g(x)}= f(x) 8(x) + g(x). dx f(x),, assuming f and g are differentiable. dx 4.12. Prove the formula dx By definition, d f(x +Ax) g (x +Ax) – f(x) g(x) {f(x) g(x)}= lim dx Ax-0 Ax f(x+Ax) {g (x +Ax) – g(x)}+g(x) {f(x+Ax) – f(x)} lim Ax-0 Ax g(x +Ax) - g(x) f(x+ Ax) - f(x) = lim f(x +Ax) + lim g(x) Ax→0 Ax Ax→0 Ax d d = f(x) 8(x)+ g(x) f(x) dx dx Another method: Let u = f(x), v = g(x). Then Au = f(x + Ax) – f(x) and A v = g(x + Ax) – g(x); i.e., f(x + Ar) = u + Au, g(x + Ar) = v + Av. Thus, d uv = lim (и + Ди)(0 + дu) — w uAv +vAu + AuAv lim dx Ar→0 Ar Ar0 Ax dv du Δυ = lim u Ar Au Au Av = u- + Ax Ar-0 Ar dx dx where it is noted that Au →0 as Ar → 0, since v is supposed differentiable and thus continuous.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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4.12) my professor says I have to explain the steps in the solved problems in the picture. Not just copy eveything down from the text.
Differentiation rules: differentiation of elementary functions d Prove the formula dx By definition, d d {f(x) g(x)}= )}= f(x)g(x)+ g(x) f(x),, assuming fand g are differentiable. 4.12. dx dx d f(x +Ax) g (x +Ax) – f(x) g(x) {f(x) g(x)}= lim dx Ax-0 Ax f(x+Ax) {g (x +Ax) – g(x)}+g(x) {f(x+Ax) – f(x)} = lim Ax-0 Ax g(x +Ax) - g(x) f(x+ Ax) – f(x) = lim f(x+Ax) + lim g(x) Ax→0 Ax Ax→0 Ax d d = f(x)8(x)+ g(x) f(x) dx dx Another method: Letu =f() , υ = g). Then Δ= f(x + Δι)-f) and Δυ- ( + Δ)-g(); ic., fr + Δε) =u+ Δι, g( + Δ)=0 + Δυ. Thus , d uv = lim (μ+ Δυ) ( + Δυ)-ω uAv +vAu + AuAv lim dx Ar-0 Ar Ar-0 Ax do du +v dx Δυ Au +v=+ Ar Δι Av = u- Ar = lim u Ar0 Ar dx where it is noted that Au → 0 as Ar → 0, since v is supposed differentiable and thus continuous.
Transcribed Image Text:Differentiation rules: differentiation of elementary functions d Prove the formula dx By definition, d d {f(x) g(x)}= )}= f(x)g(x)+ g(x) f(x),, assuming fand g are differentiable. 4.12. dx dx d f(x +Ax) g (x +Ax) – f(x) g(x) {f(x) g(x)}= lim dx Ax-0 Ax f(x+Ax) {g (x +Ax) – g(x)}+g(x) {f(x+Ax) – f(x)} = lim Ax-0 Ax g(x +Ax) - g(x) f(x+ Ax) – f(x) = lim f(x+Ax) + lim g(x) Ax→0 Ax Ax→0 Ax d d = f(x)8(x)+ g(x) f(x) dx dx Another method: Letu =f() , υ = g). Then Δ= f(x + Δι)-f) and Δυ- ( + Δ)-g(); ic., fr + Δε) =u+ Δι, g( + Δ)=0 + Δυ. Thus , d uv = lim (μ+ Δυ) ( + Δυ)-ω uAv +vAu + AuAv lim dx Ar-0 Ar Ar-0 Ax do du +v dx Δυ Au +v=+ Ar Δι Av = u- Ar = lim u Ar0 Ar dx where it is noted that Au → 0 as Ar → 0, since v is supposed differentiable and thus continuous.
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