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Use the limit laws to derive the product rule of derivatives (fg)' = f'g+ fg' by filling in the blanks: Let f and g be differentiable functions. Then: (fg)'(x) = lim definition of derivative h→0 = lim substitute (fg)(x) = f(x)g(x) h→0 - f(x)g(x+h) + = lim h→0 add & subt. f(x)g(x + h) - f(x)g(x + h) f(x)g(x+h) = lim h→0 break up fraction g(x + h) f(=) ( = lim factor h→0 = lim 9(x + h) + lim ( f(x)- using sum rule of limits h→0 h>0 lim g(x + h) \h→0 lim f(x) h→0 lim lim using product rule of limits h>0 h→0 lim + lim evaluate limits by direct sub. h→0 h>0 = f'(x)g(x) + f(x)g'(x) definition of derivatives ||

Use the limit laws to derive the product rule of derivatives (fg)' = f'g+ fg' by filling in the blanks: Let f and g be differentiable functions. Then: (fg)'(x) = lim definition of derivative h→0 = lim substitute (fg)(x) = f(x)g(x) h→0 - f(x)g(x+h) + = lim h→0 add & subt. f(x)g(x + h) - f(x)g(x + h) f(x)g(x+h) = lim h→0 break up fraction g(x + h) f(=) ( = lim factor h→0 = lim 9(x + h) + lim ( f(x)- using sum rule of limits h→0 h>0 lim g(x + h) \h→0 lim f(x) h→0 lim lim using product rule of limits h>0 h→0 lim + lim evaluate limits by direct sub. h→0 h>0 = f'(x)g(x) + f(x)g'(x) definition of derivatives ||

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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Use the limit laws to derive the product rule of derivatives (fg)' = f'g+ fg' by filling in the blanks: Let f and g be differentiable functions. Then: (fg)'(x) = lim definition of derivative h→0 = lim substitute (fg)(x) = f(x)g(x) h→0 - f(x)g(x+h) + = lim h→0 add & subt. f(x)g(x + h) - f(x)g(x + h) f(x)g(x+h) = lim h→0 break up fraction g(x + h) f(=) ( = lim factor h→0 = lim 9(x + h) + lim ( f(x)- using sum rule of limits h→0 h>0 lim g(x + h) \h→0 lim f(x) h→0 lim lim using product rule of limits h>0 h→0 lim lim evaluate limits by direct sub. h→0 h>0 = f'(x)g(x) + f(x)g'(x) definition of derivatives ||
Transcribed Image Text:Use the limit laws to derive the product rule of derivatives (fg)' = f'g+ fg' by filling in the blanks: Let f and g be differentiable functions. Then: (fg)'(x) = lim definition of derivative h→0 = lim substitute (fg)(x) = f(x)g(x) h→0 - f(x)g(x+h) + = lim h→0 add & subt. f(x)g(x + h) - f(x)g(x + h) f(x)g(x+h) = lim h→0 break up fraction g(x + h) f(=) ( = lim factor h→0 = lim 9(x + h) + lim ( f(x)- using sum rule of limits h→0 h>0 lim g(x + h) \h→0 lim f(x) h→0 lim lim using product rule of limits h>0 h→0 lim lim evaluate limits by direct sub. h→0 h>0 = f'(x)g(x) + f(x)g'(x) definition of derivatives ||
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