Partial Derivatives 14 3 Tangent Planes and Differentiability 14 4 Chain Rule 14 5 Math 241 Spring 2014 Jayadev S Athreya Spring 2014 Jayadev S Athreya Math 241 Spring 2014 Partial Derivatives 14 3 Tangent Planes and Differentiability 14 4 Chain Rule 14 5 How functions change We will be interested in how functions change and how to approximate them by functions we can understand In 1 variable the derivative played two roles Rate of change f 0 x tells us how fast the function f was changing at x Linear approximation f 0 x is the slope of the tangent line at x f x which gives us a good approximation to the function Jayadev S Athreya Math 241 Spring 2014 Partial Derivatives 14 3 Tangent Planes and Differentiability 14 4 Chain Rule 14 5 Partial derivatives Given a function of 2 variables f x y we can hold one of the variables fixed and let the other one change to get functions of one variable and can compute derivatives of those functions x The partial derivative with respect to x at a b f denoted x a b is given by if the limit exists f f a h b f a b a b lim x h h 0 Other notation Sometimes we write Dx f a b or fx a b Jayadev S Athreya Math 241 Spring 2014 Partial Derivatives 14 3 Tangent Planes and Differentiability 14 4 Chain Rule 14 5 Partial derivatives Given a function of 2 variables f x y we can hold one of the variables fixed and let the other one change to get functions of one variable and can compute derivatives of those functions y The partial derivative with respect to y at a b denoted f h a b is given by if the limit exists f f a b h f a b a b lim y h h 0 Other notation Sometimes we write Dy f a b or fy a b Jayadev S Athreya Math 241 Spring 2014 Partial Derivatives 14 3 Tangent Planes and Differentiability 14 4 Chain Rule 14 5 Example Jayadev S Athreya Math 241 Spring 2014 Partial Derivatives 14 3 Tangent Planes and Differentiability 14 4 Chain Rule 14 5 More Examples f x y ex y f x y sin xy f x y ex 2 f x y x 3 y 2 Jayadev S Athreya Math 241 Spring 2014 Partial Derivatives 14 3 Tangent Planes and Differentiability 14 4 Chain Rule 14 5 Slopes of Tangent Lines Jayadev S Athreya Math 241 Spring 2014 Partial Derivatives 14 3 Tangent Planes and Differentiability 14 4 Chain Rule 14 5 Partial derivatives 3D Given a function of 3 variables f x y z we can hold all but one of the variables fixed and let the other one change to get functions of one variable and can compute derivatives of those functions x The partial derivative with respect to x at a b c f denoted x a b c is given by if the limit exists f a h b c f a b c f a b c lim h 0 x h Other notation Sometimes we write Dx f a b c or fx a b c Jayadev S Athreya Math 241 Spring 2014 Partial Derivatives 14 3 Tangent Planes and Differentiability 14 4 Chain Rule 14 5 Partial derivatives 3D Given a function of 3 variables f x y z we can hold all but one of the variables fixed and let the other one change to get functions of one variable and can compute derivatives of those functions y The partial derivative with respect to y at a b denoted f y a b is given by if the limit exists f a b h c f a b c f a b c lim h 0 y h Other notation Sometimes we write Dy f a b c or fx a b c Jayadev S Athreya Math 241 Spring 2014 Partial Derivatives 14 3 Tangent Planes and Differentiability 14 4 Chain Rule 14 5 Partial derivatives 3D Given a function of 3 variables f x y we can hold all but one of the variables fixed and let the other one change to get functions of one variable and can compute derivatives of those functions z The partial derivative with respect to y at a b f denoted z a b is given by if the limit exists f a b c h f a b c f a b c lim h 0 z h Other notation Sometimes we write Dz f a b c or fz a b c Jayadev S Athreya Math 241 Spring 2014 Partial Derivatives 14 3 Tangent Planes and Differentiability 14 4 Chain Rule 14 5 Examples f x y z z 2 x zy xy f x y z zex 2 y f x y z sin xz cos xy f x y z ex 2 y 2 f x y z x 3 y 2 cos z Jayadev S Athreya Math 241 Spring 2014 Partial Derivatives 14 3 Tangent Planes and Differentiability 14 4 Chain Rule 14 5 Partial derivatives n dimensions Given a function of n variables f x1 xn we can hold one of the variables fixed and let the other one change to get functions of one variable and can compute derivatives of those functions xi The partial derivative with respect to xi at f p p1 pn denoted x p is given by if the limit exists f p1 pi h pn f p f p lim h 0 xi h Other notation Dxi f p or Di f p or fxi p Jayadev S Athreya Math 241 Spring 2014 Partial Derivatives 14 3 Tangent Planes and Differentiability 14 4 Chain Rule 14 5 Tangent Planes Consider the surface S R3 given by z f x y Assume Continuous partials That f has continuous partial derivatives Let P a b c be a point on S i e c f a b Jayadev S Athreya Math 241 Spring 2014 Partial Derivatives 14 3 Tangent Planes and Differentiability 14 4 Chain Rule 14 5 Cross sections Slice the surface with the planes y b To get a curve C1 C1 is given by the set of points x b f x b x R x a To get a curve C2 C2 is given by the set of points a y f a y y R Jayadev S Athreya Math 241 Spring 2014 Partial Derivatives 14 3 Tangent Planes and Differentiability 14 4 Chain Rule 14 5 Tangent Lines Let T1 Be the tangent line to C1 at P a b c The equation of the line L1 x x b f a b x a Dx f a b T2 Be the tangent line to C2 at P a b c The equation of the line will be L2 y a y f a b y b Dy f a b Jayadev S Athreya Math 241 Spring 2014 Partial Derivatives 14 3 Tangent Planes and Differentiability 14 4 Chain Rule 14 5 Tangent Lines …
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