# Chapter 3_Differentiation(3.5-3.10).pdf - u201cYou have to...

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10 “You have to apply yourself each day to becoming a little better. By becoming a little better e ach and every day, over a period of time, you will become a lot better.” – John Wooden Section 3.5: Implicit Differentiation The functions that we have met so far can be described by expressing one variable explicitly in terms of another variable. For example, 2 y x x = + and ( 2)cos y x x = + are explicit functions. In general, ( ) y f x = . However, some functions cannot be expressed as one variable in terms of another variable. In this case, we call these functions as implicit functions. For example, 2 2 2 2 2 2 2 1 2( ) 25( ) x y y or x y x y + = + = . The process of taking derivative of variable y with respect to variable x , dy dx , is called Implicit Differentiation. Ex 1 : Differentiate with respect to (w.r.t.) x . a) 4 d x dx b) 4 d y dx c) 3 5 d x y dx + Guidelines for Implicit Differentiation 1. Differentiate both sides of the equation w.r.t. x . 2. Move all terms involving dy dx on one side of the equation and collect the other terms on the other side. 3. Factor dy dx out. 4. Solve for dy dx . NOTE: dy dx can contain terms with x and y . Ex 2: Find dy dx of the following functions. a) 2 2 1 x y y + = . b) sin( ) 2 2 x y x y + = .
11 c) 2 2 sin( ) sin( ). x y x e y = Derivatives of Inverse Trigonometric Functions Ex 3: Evaluate. (a) 2 cos(arcsin ) 5 (b) 1 3 tan(cos ) 4 Now, we want to derive the derivative of 1 y cos x = , by differentiating cos y x = implicitly with respect to , 1 1 0 . x where x and y (Note: the book uses a different approach. Please also read it on page 213 214.)
12 Showing in the same fashion, we can derive the following derivatives. Ex 2: Differentiate.