Math 216 Calculus 3 Directional derivatives. Math 216 Calculus 3 Directional derivatives 1 / 6

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1 Math 216 Calculus 3 Directional derivatives Math 216 Calculus 3 Directional derivatives 1 / 6

2 How fast does f (x, y) change if you change (x, y) in the direction v The partial derivatives f x and f y measure how much a function f (x, y) changes if you start moving the input in the x-direction or the y-direction. What if you don t want to move in the x or y direction? What if instead you want to move in the direction of 2 v = 2, 2? 2 Math 216 Calculus 3 Directional derivatives 2 / 6

3 How fast does f (x, y) change if you change (x, y) in the direction v The partial derivatives f x and f y measure how much a function f (x, y) changes if you start moving the input in the x-direction or the y-direction. What if you don t want to move in the x or y direction? What if instead you want to move in the direction of 2 v = 2, 2? 2 Definition Let f (x, y) be a multivariable function. Let v = v 1, v 2 be a unit vector. The directional derivative of f (x, y) at (x 0, y 0 ) in the direction of v is D v (f ) ] f (x 0 + hv 1, y 0 + hv 2 ) f (x 0, y 0 ) (x 0, y 0 ) = lim h 0 h provided that the limit exists. Example: Let f (x, y, z) = x y z. Let v = Compute D v (f ) ] (x, y, z) 66, 6 3, 66. Math 216 Calculus 3 Directional derivatives 2 / 6

4 Directional derivatives via the chain rule Definition The directional derivative of f (x, y) in the direction of v is D v (f ) ] f (x 0 + hv 1, y 0 + hv 2 ) f (x 0, y 0 ) (x 0, y 0 ) = lim h 0 h provided that the limit exists. Notice that this can be re-expressed as a derivative with respect to h at 0. D v (f ) ] (x, y) = d dh f (x + hv 1, y + hv 2 )] h=0 Math 216 Calculus 3 Directional derivatives 3 / 6

5 Directional derivatives via the chain rule Definition The directional derivative of f (x, y) in the direction of v is D v (f ) ] f (x 0 + hv 1, y 0 + hv 2 ) f (x 0, y 0 ) (x 0, y 0 ) = lim h 0 h provided that the limit exists. Notice that this can be re-expressed as a derivative with respect to h at 0. D v (f ) ] (x, y) = d dh f (x + hv 1, y + hv 2 )] h=0 This is a composition! Math 216 Calculus 3 Directional derivatives 3 / 6

6 Directional derivatives via the chain rule Definition The directional derivative of f (x, y) in the direction of v is D v (f ) ] f (x 0 + hv 1, y 0 + hv 2 ) f (x 0, y 0 ) (x 0, y 0 ) = lim h 0 h provided that the limit exists. Notice that this can be re-expressed as a derivative with respect to h at 0. D v (f ) ] (x, y) = d dh f (x + hv 1, y + hv 2 )] h=0 This is a composition! f (x 0 + hv 1, y 0 + hv 2 ) = f (x(h), y(h)) where Math 216 Calculus 3 Directional derivatives 3 / 6

7 Directional derivatives via the chain rule Definition The directional derivative of f (x, y) in the direction of v is D v (f ) ] f (x 0 + hv 1, y 0 + hv 2 ) f (x 0, y 0 ) (x 0, y 0 ) = lim h 0 h provided that the limit exists. Notice that this can be re-expressed as a derivative with respect to h at 0. D v (f ) ] (x, y) = d dh f (x + hv 1, y + hv 2 )] h=0 This is a composition! f (x 0 + hv 1, y 0 + hv 2 ) = f (x(h), y(h)) where x(h) = x 0 + hv 1 and y(h) = y 0 + hv 2 Math 216 Calculus 3 Directional derivatives 3 / 6

8 Directional derivatives via the chain rule Definition The directional derivative of f (x, y) in the direction of v is D v (f ) ] f (x 0 + hv 1, y 0 + hv 2 ) f (x 0, y 0 ) (x 0, y 0 ) = lim h 0 h provided that the limit exists. Notice that this can be re-expressed as a derivative with respect to h at 0. D v (f ) ] (x, y) = d dh f (x + hv 1, y + hv 2 )] h=0 This is a composition! f (x 0 + hv 1, y 0 + hv 2 ) = f (x(h), y(h)) where x(h) = x 0 + hv 1 and y(h) = y 0 + hv 2 So df dh = f x x (h) + f y y (h) x (h) and y (h) are easy to compute! Math 216 Calculus 3 Directional derivatives 3 / 6

9 Directional derivatives via the chain rule Definition The directional derivative of f (x, y) in the direction of v is D v (f ) ] f (x 0 + hv 1, y 0 + hv 2 ) f (x 0, y 0 ) (x 0, y 0 ) = lim h 0 h provided that the limit exists. Notice that this can be re-expressed as a derivative with respect to h at 0. D v (f ) ] (x, y) = d dh f (x + hv 1, y + hv 2 )] h=0 This is a composition! f (x 0 + hv 1, y 0 + hv 2 ) = f (x(h), y(h)) where x(h) = x 0 + hv 1 and y(h) = y 0 + hv 2 So df dh = f x x (h) + f y y (h) x (h) and y (h) are easy to compute! df dh = f x v 1 + f y v 2. We ve proven a more user friendly formula for the directional derivative! Math 216 Calculus 3 Directional derivatives 3 / 6

10 Directional derivatives via the chain rule. Theorem Let f be a differentiable function (so that the chain rule holds). Let v be a unit vector, then D v (f ) ] (x, y) = f x (x 0, y 0 ) v 1 + f y (x 0, y 0 ) v 2 Example: Let f (x, y, z) = x y. Let 2 v = 2, 2. 2 Compute D v (f ) ] (x, y) Math 216 Calculus 3 Directional derivatives 4 / 6

11 Translating directional derivatives using the dot product: The Gradient D v (f ) ] (x, y) = f x (x 0, y 0 ) v 1 + f y (x 0, y 0 ) v 2 Notice that we can realize this as a dot product. D v (f ) ] (x, y) = Math 216 Calculus 3 Directional derivatives 5 / 6

12 Translating directional derivatives using the dot product: The Gradient D v (f ) ] (x, y) = f x (x 0, y 0 ) v 1 + f y (x 0, y 0 ) v 2 Notice that we can realize this as a dot product. D v (f ) ] (x, y) = f x (x 0, y 0 ), f y (x 0, y 0 ) v 1, v 2. Math 216 Calculus 3 Directional derivatives 5 / 6

13 Translating directional derivatives using the dot product: The Gradient D v (f ) ] (x, y) = f x (x 0, y 0 ) v 1 + f y (x 0, y 0 ) v 2 Notice that we can realize this as a dot product. D v (f ) ] (x, y) = f x (x 0, y 0 ), f y (x 0, y 0 ) v 1, v 2. In order to condense our notation, will use say gradf ] or f for the blue term in the above expression. f = gradf ] = Math 216 Calculus 3 Directional derivatives 5 / 6

14 Translating directional derivatives using the dot product: The Gradient D v (f ) ] (x, y) = f x (x 0, y 0 ) v 1 + f y (x 0, y 0 ) v 2 Notice that we can realize this as a dot product. D v (f ) ] (x, y) = f x (x 0, y 0 ), f y (x 0, y 0 ) v 1, v 2. In order to condense our notation, will use say gradf ] or f for the blue term in the above expression. f = gradf ] = f x, f y Math 216 Calculus 3 Directional derivatives 5 / 6

15 Translating directional derivatives using the dot product: The Gradient D v (f ) ] (x, y) = f x (x 0, y 0 ) v 1 + f y (x 0, y 0 ) v 2 Notice that we can realize this as a dot product. D v (f ) ] (x, y) = f x (x 0, y 0 ), f y (x 0, y 0 ) v 1, v 2. In order to condense our notation, will use say gradf ] or f for the blue term in the above expression. f = gradf ] = f x, f y We now have a prettier formula for the directional derivative. Theorem Let f be a differentiable function (so that the chain rule holds). Let v be a unit vector, then D v (f ) ] (x, y) = f v Math 216 Calculus 3 Directional derivatives 5 / 6

16 In higher dimensions Theorem Let f be a differentiable function (so that the chain rule holds). Let v be a unit vector, then D v (f ) ] (x, y) = f v Let f (x, y, z, w) = 2xyz + 2wyx + 2wxz + 2wyz. be the surface area (boundary volume?) of a 4-dimensional box. Compute f. Compute the directional derivative of f at the point (1, 1, 1, 1) in the direction v = 1/2, 1/2, 1/2, 1/2. Is f increasing or decreasing as (x, y, z, w) changes in the direction of v? Math 216 Calculus 3 Directional derivatives 6 / 6