# Unit 3 Basic Differentiationap Calculus

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Unit Guides 26 Using the Unit Guides 29 UNIT 1: Limits and Continuity 51 UNIT 2: Differentiation: Definition and Fundamental Properties 67 UNIT 3: Differentiation: Composite, Implicit, and Inverse Functions 79 UNIT 4: Contextual Applications of Differentiation 91 UNIT 5: Analytical Applications of Differentiation 109 UNIT 6: Integration. CALCULUS MAXIMUS. AP Coronavirus Calculus SCHOLARS, Tuesday, MAY 12, 2020, 1PM under a TORNADO WARNING!! Appendix: The Unit Circle. 2.3 Basic Differentiation.

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### Section 3-10 : Implicit Differentiation

For problems 1 â€“ 3 do each of the following.

1. Find (y') by solving the equation for y and differentiating directly.
2. Find (y') by implicit differentiation.
3. Check that the derivatives in (a) and (b) are the same.

1. (displaystyle frac{x}{{{y^3}}} = 1) Solution
2. ({x^2} + {y^3} = 4) Solution
3. ({x^2} + {y^2} = 2) Solution

For problems 4 â€“ 9 find (y') by implicit differentiation.

1. (2{y^3} + 4{x^2} - y = {x^6}) Solution
2. (7{y^2} + sin left( {3x} right) = 12 - {y^4}) Solution
3. ({{bf{e}}^x} - sin left( y right) = x) Solution
4. (4{x^2}{y^7} - 2x = {x^5} + 4{y^3}) Solution
5. (cos left( {{x^2} + 2y} right) + x,{{bf{e}}^{{y^{,2}}}} = 1) Solution
6. (tan left( {{x^2}{y^4}} right) = 3x + {y^2}) Solution

## Unit 3 Basic Differentiationap Calculus Calculator

For problems 10 & 11 find the equation of the tangent line at the given point.

1. ({x^4} + {y^2} = 3) at (left( {1, - sqrt 2 } right)). Solution
2. ({y^2}{{bf{e}}^{2x}} = 3y + {x^2}) at (left( {0,3} right)). Solution

## Unit 3 Basic Differentiationap Calculus 14th Edition

For problems 12 & 13 assume that (x = xleft( t right)), (y = yleft( t right)) and (z = zleft( t right)) and differentiate the given equation with respect to t.

1. ({x^2} - {y^3} + {z^4} = 1) Solution
2. ({x^2}cos left( y right) = sin left( {{y^3} + 4z} right)) Solution
AP Calculus ABâ€Ž > â€Ž

### Unit 3: Differentiation

 Day Topic 1 Definition of Derivative 2 Basic Derivative Rules 3 Product & Quotient Rules 4 Practice Power, Product & Quotient Rules 5 Chain Rule 6 Practice with Chain Rule and Derivatives of a^x 7 Implicit Differentiation 8 Some Derivatives Rules WS #1-24 9 Differentiability & Approximating Derivatives 10 Barron's Review for HW 11 Inverse Trig Derivatives 12 Derivatives of Inverse Functions (non-trig) 13 Logarithmic Differentiation 14 Review 15 Unit Exam & 'Review so Far' Packet (HW)

 Link Description Definition of Derivative Video detailing the concept of a derivative in relation to the slope of a tangent line Proof of Power Rule for Derivatives Video showing why the power rule works Derivative Proof of e^x Video explaining why the derivative of e^x = e^x Product Rule Proof Video showing a proof of the Product Rule Quotient Rule Proof Video showing a proof of the Quotient Rule Power Rule Proof Video showing a proof of the Power Rule Chain Rule Proof Video showing a proof of the Chain Rule