Implicit Differentiation in Calculus 1, Assignments of Mathematics

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CALCULUS 1

IMPLICIT DERIVATIVES

Definition

Implicit differentiation is a method for finding the derivative when one or both sides of an equation have two variables that are not easily separated. EXPLICIT IMPLICIT y = 3 x^2 + 5 x^2 + y^2 = 16 y = ! " − 10 3 x 3

• 6 xy + 2 y = 42 x = 2 sin ( θ ) + 2 cos ( θ ) sin( xy ) + cos( xy ) = x

Implicit Differentiation

Example 1. Differentiate x^2 + y^2 = 81

#" (x

#" (y

#" 81 2y #$ #" =^ - 2x %$ %$ #$ #" =^ − %" %$ =^ - " $ Example 2. Differentiate 3x^2 + y^2 = 9

#" (3x

#" (y

#" 9 6x + 2y #$ #" = 0 %$ %$ #$ #" =^ − &" %$

'" $ 2x + 2y #$ #" = 0 2y #$ #" =^ - 6x

Example 3. Differentiate y - 2x^2 + 4x = 0

#" (y)^ -^

#" (2x

#" (4x) =^

#" 0 #$ #" -^ 4x + 4 = 0 #$ #" = 4x^ -^4 #$ #" = 4(x^ -^1

1. Differentiate x^2 + y^2 + 2y = 4

#" (x

#" (y

#" (2y) =^

#" 4 2 x + 2y "# "$ + 2^ "# "$ =^0 2y "# "$ + 2^ "# "$ =^ -^2 x 2 "# "$ (y+1) =^ −^2 x "# "$ 2 (y+1) '(#()) =^ − '$ '(#()) #$ #" =-^ %" %($)!) #$ #" =^ -^ " ($)!)

Implicit Differentiation involving Trigonometric Functions

1. Differentiate sinx + 2cos2y = 1

#" sinx + 2^

#" cos2y =^

#" 1 cos x (1) + 2 [-sin2y (

#")] = 0 cos x + 2 [-sin2y ( #$ #")] = 0 4 sin2y #$ #" = cos x #$ #" =^ cos! 4 sin2y

1. Differentiate x^3 y^3 + x cos y + 8 = 0 [x^3 (3y^2 ) "# "$ + y (^3) (3x (^2) )] + [x (-sin y) "# "$ + (1) cos y + 0] = 0 "# "$ (3x (^3) y (^2) – x siny) = - 3y (^3) x (^2) - cos y "# "$ (3x^3 y^2 – x siny) (3x^3 y^2 – x siny) =^ − 3y^3 x^2 − cos y (3x^3 y^2 – x siny) "# "$ =^ − 3y^3 x^2 − cos y (3x^3 y^2 – x siny)

• Seatwork #
• 1. Differentiate y^2 + 5y + 4x^2 + 3x =
• 1. x^2 + 6xy + 5y^2 – 4x =
• 1. xy^2 - 2x^2 y =