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Write an example of a function whose derivative can be found by using the following rules: a) Product rule and special function differentiation rules b) Power rule, quotient rule, and chain rule c) Chain rule twice d) Implicit differentiation and special function differentiation rule

Write an example of a function whose derivative can be found by using the following rules: a) Product rule and special function differentiation rules b) Power rule, quotient rule, and chain rule c) Chain rule twice d) Implicit differentiation and special function differentiation rule

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Write an example of a function whose derivative can be found by using the following rules: a) Product rule and special function differentiation rules b) Power rule, quotient rule, and chain rule c) Chain rule twice d) Implicit differentiation and special function differentiation rule
Transcribed Image Text:Write an example of a function whose derivative can be found by using the following rules: a) Product rule and special function differentiation rules b) Power rule, quotient rule, and chain rule c) Chain rule twice d) Implicit differentiation and special function differentiation rule
(a) PRODUCT RULE EXAMPLES: f(x) = (3x²+4) (2x³-3x) f'(x) = (3x² + 4) £x (2x³ =3x) + x ( 3X²+4) (2x³-3x) d/dx (2x³-3x): 6x² 3! dx (3x²+4) =GX f'(x) = (3x²+4) (GX²-3) + (GX) (RX³-3x) 18x4-9x2 + 24x² -12 +12x" -18x² [ƒ¹(x) = 30 x² - 3X²-/2 PUNCTION SPECIAL A DIFFERENTIAL RULES: LOENTITIES d ax d dx d dx d dx n *² ex. x =e n-l Sinx cos X COS X = SHAX b) POWER RULE. f(x) = x² f'(x) = 2x^ f(x)=2x 21 QUOTIENT RULE: f(x)= x² + 3x xf4 EXAMPLES = |f'(x)= dt (ssint + 8cust) = (5 sint) + & (&cost) 50 sint + &at cust +-8 sint Scost -ssin t f'(x) = (x+4) (2x + 3) - (X+9x) (1) (x+4)² = 5cost 2x+3x18x+12-X-3X 2 (x+4) x² 18x+12 (x+4)2 (x+2)(x+4) (x+4)2 CHAIN RULT: f(x) = (x²+1) ³ f'(x) = 3 (x²+1 f'(x) = 6x (x²+
Transcribed Image Text:(a) PRODUCT RULE EXAMPLES: f(x) = (3x²+4) (2x³-3x) f'(x) = (3x² + 4) £x (2x³ =3x) + x ( 3X²+4) (2x³-3x) d/dx (2x³-3x): 6x² 3! dx (3x²+4) =GX f'(x) = (3x²+4) (GX²-3) + (GX) (RX³-3x) 18x4-9x2 + 24x² -12 +12x" -18x² [ƒ¹(x) = 30 x² - 3X²-/2 PUNCTION SPECIAL A DIFFERENTIAL RULES: LOENTITIES d ax d dx d dx d dx n *² ex. x =e n-l Sinx cos X COS X = SHAX b) POWER RULE. f(x) = x² f'(x) = 2x^ f(x)=2x 21 QUOTIENT RULE: f(x)= x² + 3x xf4 EXAMPLES = |f'(x)= dt (ssint + 8cust) = (5 sint) + & (&cost) 50 sint + &at cust +-8 sint Scost -ssin t f'(x) = (x+4) (2x + 3) - (X+9x) (1) (x+4)² = 5cost 2x+3x18x+12-X-3X 2 (x+4) x² 18x+12 (x+4)2 (x+2)(x+4) (x+4)2 CHAIN RULT: f(x) = (x²+1) ³ f'(x) = 3 (x²+1 f'(x) = 6x (x²+
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