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# MATH 1760-003 Math 1760

Toledo

GPA 3.96

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This 7 page Class Notes was uploaded by Amalachi Notetaker on Tuesday August 30, 2016. The Class Notes belongs to Math 1760 at University of Toledo taught by Susan Vayo in Fall 2016. Since its upload, it has received 24 views. For similar materials see Calculus 2 App in Natural Sciences and Mathematics at University of Toledo.

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Date Created: 08/30/16

Derivative Rules: f(x+h − f (x) lim Definition: h→0 h d Notation: Derivative = y’ = f’(x) = dx f(x) = [f(x)]’ d d 1. dx c)=0 dx (7) = 0 d d 3 3 −1 n n-1 4 4 2. dx ( x ) = n x dx ( x ) = 4 x d ∙ ∙ d 5 ∙ 4 3. dx (a f(x)) = a f'(x) dx (3x ) = 3 5x = 15x 4 d ± ± d 3 4. dx [ f(x) g(x)] = f’(x) g’(x) dx (2x + 4x) = 6x + 4 f(x)∙g(x) 5. d ] = f’(x) ∙ g(x) + f(x) ∙ g’(x) dx¿ d [3x (4x + 1)] = 6x(4x + 1) + 3x (4) = 36x + 6x 2 dx d f (x) f (x)∙gx − f2(x)∙g’(x) d 2 x+5 6. dx [ g(x) ] = (g(x)) dx [ 7 x−3 ] = 2(7x−3 −(2x+5)(7) (7x−3 )2 7. Chain Rule: y = (fοg)(x) y’ = f’[g(x)] ∙ g(x) 5 4 y = (3x +8) y’ = 5(3x +8) (3) 8. Trigonometric Functions: d d (a) dx (sin x) = cos x (d) dx (cot x) = - csc x2 d d (b) dx (cos x) = - sin x (e) dx (sec x) = sec x tan x d d (c) (tan x) = sec x 2 (f) dx dx (csc x) = - csc x cot x 9. Exponential Functions: d du d (a) dx (e ) = e u ∙ dx dx (e ) = e 2x ∙ 2 d u u ∙ du ∙ d (b) dx (a ) = a dx ln a dx 4x 4x ∙ ∙ (3 ) = 3 4 ln 3 10. Logarithmic Functions: d 1 du (a) (ln u) = ∙ dx u dx d 1 (ln (3x – 2x) = 3 ∙ (9x -2) dx 3x −2x d log u 1 ∙ du ∙ 1 (b) dx ( a ) = u dx lna lo5 2x d 1 ∙ 1 ¿ ) = 2 x (2) ln5 dx 11. Inverse Trigonometric Functions: d sin u 1 ∙ du (a) dx ( ) = √1−u 2 dx sin y = u d 1 du cos u 2 ∙ (b) dx ( ) = - √ 1−u dx d 1 du d (c) dx (tan u) = 1+u2 ∙dx dx (tan -1 1 3x ) = 1+9x 4 ∙ (6x) Derivative of an inverse function: 1 dy dx d -1 1 dx = or dx (f (x)) = f ( f( )) dy y = sin x-1 → sin y = x dx cos y = dy U 1 u uuuu x uu u √ 1−x2 1 1 dy 1 dy ∙ dx = cosy = √1−x 2 or in general dx = √1−u 2 du dx when y = sin u -1 Y = sin 4x y’ = (cos 4x)4= 4 cos 4x 2 2 2 2 2 Y = tan 3x y’ =( sec 3x )6x = 6x sec 3x 3 3 3 e x ex ex Y = (sec 2x) y’ = (sec 2x)’ + (sec 2x)( )’ x Y’ = (sec 2x tan 2x)(2 ¿e + (sec 2x) ex3 2 (3x ) F’(x) = 2x 4 F(x) = ?

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