14.3: Partial Derivatives, 14.4: The Chain Rule, and 14.5: Directional Derivatives and Gradient Vectors

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1 14.3: Partial Derivatives, 14.4: The Chain Rule, and 14.5: Directional Derivatives and Gradient Vectors TA: Sam Fleischer November 10 Section 14.3: Partial Derivatives Definition: Partial Derivative with respect to x, y The partial derivative of f(x, y) with respect to x at the point (x 0, y 0 ) is f x f d x dx f(x, y 0) f(x 0 + h, y 0 ) f(x 0, y 0 ) lim h 0 (x0,y 0 ) h (x0,y 0 ) The partial derivative of f(x, y) with respect to y at the point (x 0, y 0 ) is f y f y d (x0,y 0 ) dy f(x 0, x) f(x 0, y 0 + h) f(x 0, y 0 ) lim h 0 (x0,y 0 ) h Example: Taking Partial Derivatives f(x, y) y sin (xy) To find f x (the partial derivative of f with respect to x), treat y as a constant. f x y cos (xy)y y cos (xy) To find f y (the partial derivative of f with respect to y), treat x as a constant. f y (1) sin (xy) + y cos (xy)x sin (xy) + xy cos (xy) Example: Implicit Differentiation Find z x if the equation yz ln z x + y defines z as a function of two independent variables x and y and the partial derivative exists. d d (yz ln z) (x + y) dx dx 1

2 y z x 1 z z x ( y 1 ) z z x 1 z x Definition: Second-Order Partial Derivatives x yz 1 When we differentiate a function f(x, y) twice, we produce its second-order derivatives. Notation: f x or f xx, f x y or f yx, and f y or f yy f y x f xy The defining equations are f x ( ) f, x x f x y ( ) f x y f Note that f yx means you first take the derivative with respect to y, and then take the x y derivate with respect to x. In general, f xy f yx. Theorem : Mixed Derivative Theorem If f(x, y) and its partial derivatives f x, f y, f xy, and f yx are defined throughout an open region containing a point (a, b) and are all continuous at (a, b), then f xy (a, b) f yx (a, b) Example: Partial Derivatives of Higher Order Find f yxyz if f(x, y, z) 1 xy z + x y f(x, y, z) 1 xy z + x y f y 4xyz + x f yx 4yz + x f yxy 4z f yxyz 4

3 Definition: Differentiability A function z f(x, y) is differentiable at (x 0, y ) if f x (x 0, y 0 ) and f y (x 0, y 0 ) exists and z satisfies an equation of the form z f x (x 0, y 0 ) x + f y (x 0, y 0 ) y + ɛ 1 x + ɛ y in which each of ɛ 1, ɛ 0 as both x, y 0. We call f differentiable if it is differentiable at every point in its domain, and say that its graph is a smooth surface. Theorem 3: The Increment Theorem for Functions of Two Variables Suppose that the first partial derivatives of f(x, y) are defined throughout an open region R containing the point (x 0, y 0 ) and that f x and f y are continuous at (x 0, y 0 ). Then the change z f(x 0 + x, y 0 + y) f(x 0, y 0 ) in the value of f that results from moving from (x 0, y 0 ) to another point (x 0 + x, y 0 + y) in R satisfies an equation of the form z f x (x 0, y 0 ) x + f y (x 0, y 0 ) y + ɛ 1 x + ɛ y in which each of ɛ 1, ɛ 0 as both x, y 0. Corollary of Theorem 3 If the partial derivatives f x and f y of a function f(xy) are continuous throughout an open region R, then f is differentiable at every point of R. Theorem 4 - DIfferentiability Implies Continuity If a function f(x, y) is differentiable at (x 0, y 0 ), then f is continuous at (x 0, y 0 ). Section 14.4: The Chain Rule Theorem 5: Chain Rule for Functions of One Independent Variable and Two Intermediate Variables If w f(x, y) is differentiable and if x x(t), y y(t) are differentiable functions of t, then the composite w f(x(t), y(t)) is a differentiable function of t and or dt f x(x(t), y(t)) x (t) + f y (x(t), y(t)) y (t) dt f dx x dt + f dy y dt 3

4 Theorem 6: Chain Rule for Functions of One Independent Variable and Three Intermediate Variables If w f(x, y, z) is differentiable and x, y, and z are differentiable functions of t, then w is a differentiable function of t and Example dt dx x dt + dy y dt + dz z dt Find dt if w xy + z, x cos t, y sin t, z t Use Theorem 6: dt dx x dt + dy y dt + dz z dt (y)( sin t) + (x)(cos t) + (1)(1) (sin t)( sin t) + (cos t)(cos t) + 1 sin t + cos t + 1 cos t + 1 What is the derivative at t 0? ( ) 1 + cos 0 dt t0 Theorem 7: Chain Rule for Two Independent Variables and Three Intermediate Variables Suppose that w f(x, y, z), x g(r, s), y h(r, s), and z k(r, s). If all four functions are differentiable, then w has partial derivatives with respect to r and s given by the formulas r x x r + y y r + z z r s x x s + y y s + z z s Theorem 8: A Formula for Implicit Differntiation Suppose that F (x, y) is differentiable and that the equation F (x, y) 0 defines y as a differentiable function of x. Then at any point where F y 0, dy dx F x F y 4

5 Example Find dy dx if y x sin xy 0. dy dx F x F y x y cos xy y x cos xy x + y cos xy y x cos xy Expansion of Theorem 8 to Three Variables Suppose F (x, y, z) 0 and z f(x, y). Assuming F and f are differentiable functions, then z x F x F z and z y F y F z Expansion of Chain Rule to Functions of n variables In general, suppose z f(x 1, x,..., x n ) is a differential function of the intermediate variables x 1, x,..., x n where n is a positive integer (n N). Also suppose each x i is a differentiable function of the independent variables t 1, t,..., t m, with m N. In equation form, x 1 g 1 (t 1, t,..., t m ) x g (t 1, t,..., t m ). x n g n (t 1, t,..., t m ) Then w is a differential function of each of the independent variables t 1, t,..., t m, and the partial derivatives of w with respect to each t i are More compactly, x 1 + x + + x n t 1 x 1 t 1 x t 1 x n t 1 x 1 + x + + x n t x 1 t x t x n t. x 1 + x + + x n t m x 1 t m x t m x n t m i1 i1 i1 t 1 t t m t j i1 t j for j 1,,..., m 5

6 Directional Derivatives and Gradient Vectors Definition: Directional Derivative The derivative of f at P 0 (x 0, y 0 ) in the direction of the unit vector u u 1 i u j is the number ( ) df f(x 0 + su 1, y 0 + su ) f(x 0, y 0 ) lim ds s 0 u,p 0 s provided the limit exists. The directional derivative is also denoted ( ) df (D u f) ds P0 u,p 0 and is read The derivative of f at P 0 in the direction of u. Definition: Gradient Vector The gradient vector (gradient) of f(x, y) at a point P is the vector f f + f y j Theorem 9: The Directional Derivative is a Dot Product If f(x, y) is differentiable in an open region containing P 0 (x 0, y 0 ), then (D u f) P0 ( f) P0 u In words, the derivative of f at P 0 in the direction of u is the dot product of the gradient f at P 0 and u. In brief, Example (D u )f f u Find the derivative of f(x, y) xe y + cos xy at the point (, 0) in the direction of v 3i 4j. First, find the unit direction vector u v v v i 4 5 j Then we need to find the partial derivatives of f at (, 0) because together, they make up the gradient, f. f x e y y sin xy f x (, 0) e 0 0 sin( 0) f y xe y x sin xy f y (, 0) e 0 sin( 0) 0 6

7 Plug these values into the definition of gradient. f (,0) f x (, 0)i + f y (, 0)j i + j Then the directional derivative of f at (, 0) in the direction of u is Properties of the Directional Derivative (D u f) (,0) f (,0) u ( 3 5 i 4 ) 5 j (i + j) The function f increases most rapidly when cos θ 1, i.e. when θ 0, i.e. when u is the direction of f. That is, at each point P in its domain, f increases most rapidly in the direction of the gradient vector f at P. The derivative in this direction is D u f f cos 0 f. Similarly, f decreases most rapidly in the direction of f. The derivative in this direction is D u f f cos π f 3. Any direction u orthogonal to a gradient f 0 is adirection of zero change in f because θ π and Example D u f f cos π f 0 0 Let f(x, y) x + y, and consider the point (1, 1). The function increases most rapidly in the direction of f. The unit vector of ( f) (1,1) is ( f) xi + yj ( f) (1,1) i + j u i + j 1 i + 1 j The function decreases most rapidly in the direction ( f) (1,1) 1 i 1 j The directions of zero change at (1, 1) are the directions ofthogonal to f: n 1 i + 1 j and n 1 i 1 j 7

8 Important Concept At every point (x 0, y 0 ) in the domain of a differentiable function f(x, y), the gradient of f is normal to the level curve through (x 0, y 0 ). Tangent Line to a Level Curve f x (x 0, y 0 )(x x 0 ) + f y (x 0, y 0 )(y y 0 ) 0 Notice this is the same as point-slope form from elementary algebra. y y 0 m(x x 0 ) where by Theorem 8. m f x f y dy dx Algebra Rules for Gradients 1. Sum Rule: (f + g) f + g. Difference Rule: (f g) f g 3. Constant Multiple Rule: (kf) k f 4. Product Rule: (fg) f g + g f ( ) f g f f g 5. Quotient Rule: g g Gradients of Functions of n variables For a differential function f(x 1, x,... x n ) and a unit vector u u 1, u,..., u n in space, we have f f, f,..., f x 1 x x n and The Derivative Along a Path D u f f u f x 1 u 1 + f x u +... f x n u n i1 f u i Let r(t) x(t)i + y(t)j + z(t)k be a smooth path C and w f(r(t)) a scalar function along C. Then dt dx x dt + dy y dt + dz z dt or in vector notation, d dt f(u(t)) f(r(t)) r (t) 8