CALCULUS Trancedental Functions. Chain Rule 1 ... - La Citadelle

CALCULUS1. Differentiate: f(x) = 10 2−3xTrancedental Functions. Chain Rule2. Differentiate: f(x) = 10 4+2x3. Differentiate: f(x) = log(5 − 3x − 2x 2 − 4x 3 )4. Differentiate: f(x) = log 4 (−1 + 4x)5. Differentiate: f(x) = sin(−1 − 4x − x 2 + 3x 3 )6. Differentiate: f(x) = 3 −4x+5x2 −2x 37. Differentiate: f(x) = log 5 (−5 − 3x + 4x 2 − 4x 3 )8. Differentiate: f(x) = ln(−1 + x + x 2 )9. Differentiate: f(x) = ln(−5 − 5x − x 3 )10. Differentiate: f(x) = log(5 − 5x − 5x 2 )Answers: 1. f ′ (x) = (−3)(ln 10)10 2−3x2. f ′ (x) = (2)(ln 10)10 4+2x3. f ′ (x) =4. f ′ (x) =−3 − 4x − 12x 2(ln 10)(5 − 3x − 2x 2 − 4x 3 )4(ln 4)(−1 + 4x)5. f ′ (x) = (−4 − 2x + 9x 2 ) cos(−1 − 4x − x 2 + 3x 3 )6. f ′ (x) = (−4 + 10x − 6x 2 )(ln 3)3 −4x+5x2 −2x 37. f ′ (x) =8. f ′ (x) =9. f ′ (x) =10. f ′ (x) =−3 + 8x − 12x 2(ln 5)(−5 − 3x + 4x 2 − 4x 3 )1 + 2x−1 + x + x 2−5 − 3x2−5 − 5x − x 3−5 − 10x(ln 10)(5 − 5x − 5x 2 )c○ 2009 La Citadelle 1 of 3 www.la-citadelle.com

CALCULUSSolutions:1. f ′ (x) = ddx f(x) = ddx 102−3x◭ Apply:= ( (ln 10)10 2−3x) ddx (2 − 3x) ◭ Apply: ddx xn = nx n−1= ( (ln 10)10 2−3x) (−3) ◭ Simplify, if necessary.∴ ddx 102−3x = (−3)(ln 10)10 2−3x2. f ′ (x) = ddx f(x) = ddx 104+2x◭ Apply:= ( (ln 10)10 4+2x) ddx (4 + 2x) ◭ Apply: ddx xn = nx n−1= ( (ln 10)10 4+2x) (2) ◭ Simplify, if necessary.∴ ddx 104+2x = (2)(ln 10)10 4+2xddx 10f(x) = (ln 10)10 f(x) f ′ (x)ddx 10f(x) = (ln 10)10 f(x) f ′ (x)Trancedental Functions. Chain Ruleddx log f(x) = 1(ln 10)f(x) f ′ (x)3. f ′ (x) = ddx f(x) = ddx log(5 − 3x − 2x2 − 4x 3 ) ◭ Apply:()1d=(ln 10)(5 − 3x − 2x 2 − 4x 3 ) dx (5 − 3x − 2x2 − 4x 3 ) ◭ Apply:()1=(ln 10)(5 − 3x − 2x 2 − 4x 3 (−3 − 4x − 12x 2 ) ◭ Simplify, if necessary.)∴ ddx log(5 − 3x − 2x2 − 4x 3 −3 − 4x − 12x 2) =(ln 10)(5 − 3x − 2x 2 − 4x 3 )4. f ′ (x) = ddx f(x) = ddx log 4(−1 + 4x)()1 d=(ln 4)(−1 + 4x)(1=(ln 4)(−1 + 4x)∴ ddx log 44(−1 + 4x) =(ln 4)(−1 + 4x)◭ Apply:dx (−1 + 4x) ◭ Apply: ddx xn = nx n−1)(4) ◭ Simplify, if necessary.5. f ′ (x) = ddx f(x) = ddx sin(−1 − 4x − x2 + 3x 3 )= ( cos(−1 − 4x − x 2 + 3x 3 ) ) ddx (−1 − 4x − x2 + 3x 3 )ddx xn = nx n−1ddx log 1b f(x) =(ln b)f(x) f ′ (x)d◭ Apply:dx sin f(x) = (cos f(x))f ′ (x)d◭ Apply:dx xn = nx n−1= ( cos(−1 − 4x − x 2 + 3x 3 ) ) (−4 − 2x + 9x 2 ) ◭ Simplify, if necessary.∴ ddx sin(−1 − 4x − x2 + 3x 3 ) = (−4 − 2x + 9x 2 ) cos(−1 − 4x − x 2 + 3x 3 )6. f ′ (x) = ddx f(x) = d−2x 3dx 3−4x+5x2(= (ln 3)3 −4x+5x2 −2x 3) ddx (−4x + 5x2 − 2x 3 )(= (ln 3)3 −4x+5x2 −2x 3) (−4 + 10x − 6x 2 )◭ Apply:ddx bf(x) = (ln b)b f(x) f ′ (x)◭ Apply:ddx xn = nx n−1◭ Simplify, if necessary.c○ 2009 La Citadelle 2 of 3 www.la-citadelle.com

CALCULUS∴ ddx 3−4x+5x2 −2x 3 = (−4 + 10x − 6x 2 )(ln 3)3 −4x+5x2 −2x 37. f ′ (x) = ddx f(x) = ddx log 5(−5 − 3x + 4x 2 − 4x 3 d) ◭ Apply:dx log b f(x) =()1d=(ln 5)(−5 − 3x + 4x 2 − 4x 3 ) dx (−5 − 3x + 4x2 − 4x 3 ) ◭ Apply:()1=(ln 5)(−5 − 3x + 4x 2 − 4x 3 (−3 + 8x − 12x 2 ) ◭ Simplify, if necessary.)∴ ddx log 5(−5 − 3x + 4x 2 − 4x 3 −3 + 8x − 12x 2) =(ln 5)(−5 − 3x + 4x 2 − 4x 3 )8. f ′ (x) = ddx f(x) = ddx ln(−1 + x + x2 ) ◭ Apply:()1 d=−1 + x + x 2 dx (−1 + x + x2 ) ◭ Apply:()1=−1 + x + x 2 (1 + 2x) ◭ Simplify, if necessary.∴ ddx ln(−1 + x + 1 + 2xx2 ) =−1 + x + x 2Trancedental Functions. Chain Ruled1ln f(x) =dx f(x) f ′ (x)ddx xn = nx n−19. f ′ (x) = ddx f(x) = ddx ln(−5 − 5x − x3 ) ◭ Apply:()1 d=−5 − 5x − x 3 dx (−5 − 5x − x3 ) ◭ Apply:()1=−5 − 5x − x 3 (−5 − 3x 2 ) ◭ Simplify, if necessary.∴ ddx ln(−5 − 5x − −5 − x3 3x2) =−5 − 5x − x 310. f ′ (x) = ddx f(x) = ddx log(5 − 5x − 5x2 )()1d=(ln 10)(5 − 5x − 5x 2 ) dx (5 − 5x − 5x2 )(1=(ln 10)(5 − 5x − 5x 2 )∴ ddx log(5 − 5x − −5 − 10x5x2 ) =(ln 10)(5 − 5x − 5x 2 )◭ Apply:d1ln f(x) =dx f(x) f ′ (x)ddx xn = nx n−1◭ Apply:1(ln b)f(x) f ′ (x)ddx xn = nx n−1ddx log f(x) = 1(ln 10)f(x) f ′ (x)ddx xn = nx n−1)(−5 − 10x) ◭ Simplify, if necessary.c○ 2009 La Citadelle 3 of 3 www.la-citadelle.com