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Suppose that the derivatives f'(z) and g'(z) exist. Show: f(z)g(z)-g' (z)f(z) g²(z) b) [f(g(z))] = f(g(z))g (z). a)

Suppose that the derivatives f'(z) and g'(z) exist. Show: f(z)g(z)-g' (z)f(z) g²(z) b) [f(g(z))] = f(g(z))g (z). a)

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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SOLVE STEP BY STEP IN DIGITAL FORMAT ÿ ÿ¾Ü Ð ð ♥ing @@ !! ?? !! ??! ¿¡ !? W X & a i √√√XXXXXOOO O ☐☐ Suppose that the derivatives f'(z) and g'(z) exist. Show: a) [12] = f (2)g(2)-g′(2)ƒ (2) g²(z) b) [f(g(z))] = f(g(z))g'(z). " DOOO ♫♬ bh #°Ø ♡ □ ☨ + + ♀ 未 X + D *** <R O o °C °F ► AVADA D A PO + V 4 ✓ P *
Transcribed Image Text:SOLVE STEP BY STEP IN DIGITAL FORMAT ÿ ÿ¾Ü Ð ð ♥ing @@ !! ?? !! ??! ¿¡ !? W X & a i √√√XXXXXOOO O ☐☐ Suppose that the derivatives f'(z) and g'(z) exist. Show: a) [12] = f (2)g(2)-g′(2)ƒ (2) g²(z) b) [f(g(z))] = f(g(z))g'(z). " DOOO ♫♬ bh #°Ø ♡ □ ☨ + + ♀ 未 X + D *** <R O o °C °F ► AVADA D A PO + V 4 ✓ P *
Expert Solution
Step 1

Let f(z) be a differentiable function. Then the derivative of a function is defined as f'(z)=limh→0f(z+h)-f(z)h, provided that this limit exists.

 

a)

Suppose that f'(z),g'(z) exits.

Then

f'(z)=limh→0f(z+h)-f(z)h and g'(z)=limh→0g(z+h)-g(z)h

To prove f(z)g(z)'=f'(z)g(z)-g'(z)f(z)(g(z))2

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