Y dx dx step 4 solve for dy dx step 5 express the

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Unformatted text preview: = sec 2 sec e sin ( x ) ⋅ sec e sin ( x ) tan e sin ( x ) ⋅ e sin ( x ) (( y ′ = sec sec e 2 sin( x ) ))⋅ [sec(e sin( x ) )tan(e ln ( f ( x )) by using the sin( x ) )]⋅ [e ( d sin ( x ) e dx d (sin (x )) dx sin( x ) cos( x) ) 1 dy d ⋅ = (ln f ( x )) ). y dx dx Step 4. Solve for dy dx . (thus: Step 5. Express the answer in terms of x only (substitution f(x) for y). EXAMPLE: Find the derivative of the function y = x x . Take the natural logarithm of both d u(x) du c = c u ( x ) ⋅ ln c ⋅ dx dx sides: ln Find the derivative of the function y = x ln x . Differentiate both sides: EXAMPLE: x 1 dy 1 −1 / 2 ⎛ ln x ⎞ =x = x −1 / 2 ⎜1 + ln x + ⎟ y dx 2 x 2⎠ ⎝ y = 7x SOLUTION: dy x ⎛ ln x ⎞ Solve for dy : = ⎜1 + ⎟ dx dx 2⎠ x⎝ x y ′ = 7 x ln 7 ⋅ d (x ) dx y ′ = 7 x ln 7 Applications of the Derivative HIGHER ORDER DERIVATIVES: Critical points: The values of x ∈ domain of f(x) such that f ′ x = 0 or f ′( x) is not defined. f ′′( x) = d ⎛ dy ⎞ . ⎜⎟ () dx ⎝ dx ⎠ f ′( p ) = 0 TECHNIQUES OF DIFFERENTIATION First derivative test: If Implicit differentiation: to differentiate an implicit f...
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This note was uploaded on 03/15/2010 for the course MAT MAT135 taught by Professor Treung during the Fall '08 term at University of Toronto- Toronto.

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