Unit 10 Differentiation | University of California, Berkeley

University:
University of California, Berkeley
Course:
MATH W54 | Linear Algebra and Differential Equations
Academic year:

2021
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464

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4
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96XGbl6o

Unit 10 Differentiation ● Definition of the Derivative of a Function ○ 𝑓′(𝑥) = 𝑙𝑖𝑚 𝑓(𝑥+𝛥𝑥)−𝑓(𝑥) 𝛥𝑥 𝛥𝑥→0 ● Slope = 𝑙𝑖𝑚 𝛥𝑥→0 𝑓(𝑥+𝛥𝑥)−𝑓(𝑥) 𝑥+𝛥𝑥−𝑥 ● Derivative Notations: ○ 𝑓′(𝑥) =first derivative of f ○ 𝑑𝑦 = 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑑𝑥 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 = The derivative of y with respect to x ○ 𝑦′= “y” prime ○ 𝑑 𝑑𝑥 [𝑓(𝑥)]= take the derivative of f(x) with respect to x ● Example: Find the derivative of 𝑓(𝑥) = 𝑥 2 ● Alternative form of the derivative (at a given point) (𝑐, 𝑓(𝑐)) ○ 𝑓′(𝑥) = 𝑙𝑖𝑚 𝑥→𝑐 𝑓(𝑥)−𝑓(𝑐) 𝑥−𝑐 2 ● Example: Find the derivative with respect to t for the function 𝑦 = 𝑡 ,when t = 2 ● Differentiability and Continuity ○ Theorem: If f is differentiable at x = c, then f is continuous at x = c ● 4 Equations for differentiable equations 1. Not continuous 2. Sharp Turns/Cusps (absolute value) 3. Vertical tangent line 4. End Points Basic Differentiation Rules ● The Constant Rule ○ 𝑓′(𝑥) = 0, 𝑐 = # ● Power Rule ○ 𝑓′(𝑥 𝑛 ) = 𝑛𝑥 𝑥−1 ● Constant Multiple Rule ○ 𝑓′(𝑐𝑓(𝑥)) = 𝑐𝑓′(𝑥) ● Sum and Difference Rules ○ 𝑓′[𝑓(𝑥) ± 𝑔(𝑥)] = 𝑓′(𝑥) ± 𝑔′(𝑥) ● Product Rule ○ 𝑓′[𝑓(𝑥)𝑔(𝑥)] = 𝑓(𝑥)𝑔′(𝑥) + 𝑓′(𝑥)𝑔(𝑥) ● Quotient Rule 𝑓(𝑥) ○ 𝑓′(𝑥)[𝑔(𝑥)] = 𝑔(𝑥)𝑓′(𝑥)−𝑓(𝑥)𝑔′(𝑥) 𝑔2 (𝑥) ● Chain Rule ○ 𝑓′[𝑓(𝑔(𝑥))] = 𝑓′(𝑔(𝑥)) ⋅ 𝑔′(𝑥) ● Derivatives of Trigonometric Functions ○ ○ ○ ○ ○ ○ 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑠𝑖𝑛𝑥 = 𝑐𝑜𝑠𝑥 𝑐𝑜𝑠𝑥 = −𝑠𝑖𝑛𝑥 𝑡𝑎𝑛𝑥 = 𝑠𝑒𝑐 2 𝑥 𝑐𝑜𝑡𝑥 = −𝑐𝑠𝑐 2 𝑥 𝑠𝑒𝑐𝑥 = 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥 𝑐𝑠𝑐𝑥 = −𝑐𝑠𝑐𝑥𝑐𝑜𝑡𝑥 Rate of Change ● 𝑟𝑎𝑡𝑒 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑖𝑚𝑒 ● 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡𝑖𝑚𝑒 1 ● Position Function: 𝑠(𝑡) = − 2 𝑎𝑡 2 + 𝑣0 𝑡 + 𝑠0 ○ a = acceleration (gravity) ○ v0 = initial velocity ○ s0 = Initial position ● The sign on velocities indicates the direction. ● Instantaneous Velocity is the velocity at a single point. 1 ○ 𝑠(𝑡) = − 2 𝑎𝑡 2 + 𝑣0 𝑡 + 𝑠0 Implicit Differentiation ● Implicit: Not solved for dependent variable ● Explicit: Solved for dependent variable Related Rates 𝑑 ● Find 𝑑𝑡 [𝑥 2 + 3] 𝑑𝑥 ○ 2𝑥 ⋅ 𝑑𝑡 + 0 𝑑 𝜋 ● Find 𝑑𝑡 [ 3 𝑟 2 ℎ] ○ 𝜋 3 𝑑ℎ 𝑟 2 ⋅ 𝑑𝑡 + 2𝜋 3 𝑑𝑟 𝑟ℎ ⋅ 𝑑𝑡