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Recall : Explicit equation Implicit equation
𝒚 ൌ 𝒙 𝟏𝟔 𝒙𝟐^ 𝒚𝟐^ ൌ 𝟏𝟔
Chain Rule and Implicit Differentiation
In terms of 𝑥 In terms of 𝑦
ௗ௫^ ௗ 𝑥 ൌ^ ௗ௫ௗ௫ ൌ^1 ௗ௫ௗ 𝑦 ൌ^ ௗ௬ௗ௫
ௗ௫^ ௗ 𝑥^ ଶ^ ൌ^2 𝑥^ ௗ௫ௗ௫ ൌ^2 𝑥^ ௗ௫ௗ 𝑦^ ଶ^ ൌ^2 𝑦^ ௗ௬ௗ௫
ௗ௫^ ௗ 𝑒^ ହ௫^ ൌ 𝑒^ ହ௫^5 ௗ௫ௗ௫ ൌ 𝑒^ ହ௫^5 ௗ௫ௗ 𝑒^ ହ௬^ ൌ 𝑒^ ହ௬^5 ௗ௫ௗ௫
Implicit Differentiation Example: Find ௗ௬ௗ௫ for 𝑦 ଶ^ െ 5 𝑥 ଷ^ ൌ 3 𝑦
Step 1: Take the derivative. Each time the derivative of “y” is involved, include a ௗ௬ௗ௫.
Step 2: Gather all terms with ௗ௬ௗ௫ on the left side, everything else on the right.
Step 3: Factor out the ௗ௬ௗ௫ if necessary, to create only one ௗ௬ௗ௫ term.
Step 4. Solve for ௗ௬ௗ௫.
1. 𝑦 ଷ^ െ 2 𝑥 ൌ 𝑥 ସ^ 2 𝑦 2. sinሺ𝑥𝑦ሻ ൌ 10 𝑥
3.2 Implicit Differentiation
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Calculus Notes
Derivative 3. Find the equation of all tangent lines for at a point – implicit differentiation 𝑥 ଶ (^) 𝑦 ଶ (^) ൌ. 4 when 𝑥 ൌ 1.
Horizontal and Vertical Tangent Lines Horizontal tangent lines exist when the slope, ௗ௬ௗ௫ ൌ 𝟎. Vertical tangent lines exist when the slope, ௗ௬ௗ௫ is undefined.
4. graphFind all 3 𝑥 horizontal ଶ 2 𝑦 ଶ ൌ tangent lines of the 16. 5. graphFind all 3 𝑥 vertical ଶ 2 𝑦 ଶ tangent lines of the ൌ 16.
3.2 Calculus Implicit Differentiation Find 1. 5 (^) 𝑥ௗ௬ௗ௫ ଶ. 2 𝑦 ଷ (^) ൌ 4 2. 5 𝑦 ଶ (^) 3 ൌ 𝑥 ଶ (^) 3. sinሺ𝑥 𝑦ሻ ൌ 2 𝑥
Practice
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x
y
Find the slope of the tangent line at the given point. Show work. 14. 2 ൌ 3 𝑥 ସ (^) 𝑥𝑦 ସ (^) at ሺെ1, 1ሻ 15. 𝑥 ln 𝑦 ൌ 4 െ 2 𝑥 at ሺ2, 1ሻ
Find the equation of the tangent line at the given point. 16. 𝑥 ଶ (^) 𝑦 ଶ (^) 19 ൌ 2 𝑥 12 𝑦 at ሺ4, 3ሻ (^) 17. 𝑥 sin 2𝑦 ൌ 𝑦 cos 2𝑥 at ቀ గ ସ ,^ గଶ ቁ
Find the equations of all horizontal and vertical tangent lines. Calculator allowed. Round to three decimals. 18. 𝑥 ଶ 𝑥 2 𝑦 ଶ ൌ 8
Horizontal: _________________________________ Vertical:____________________________________
Horizontal: _________________________________ Vertical:____________________________________
20. Find the slope of the normal line to 𝑦 ൌ 𝑥 cosሺ𝑥𝑦ሻ at ሺ0,1ሻ.
(A) 1 (B) െ 1 (C) 0 (D) 2 (E) Undefined
21. The graph of 𝑓ሺ𝑥ሻ, shown below, consists of a semicircle and two-line segments. 𝑓 ᇱ^ ሺ 1 ሻ ൌ
(A) െ 1 (B) െ √ଷଵ (C) √ଷଵ (D) 1 (E) √ 3
22. Find the value(s) of ௗ௬ௗ௫ of 𝑥 ଶ^ 𝑦 𝑦 ଶ^ ൌ 5 at 𝑦 ൌ 1.
(A) െ ଷଶ only (B) െ ଶଷ only (C) ଶଷ only (D) േ ଶଷ (E) േ ଷଶ
(^)
x
3.2 Implicit Differentiation^ Test Prep
