Differentials, Chainr Rule and Directional Derivatives
Differentials, Chainr Rule and Directional Derivatives Math 2419
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This 3 page Class Notes was uploaded by Saul Cervantes on Monday October 12, 2015. The Class Notes belongs to Math 2419 at University of Texas at Dallas taught by Anotoly Ezlydon in Fall 2015. Since its upload, it has received 44 views. For similar materials see Accelerated Calculus II in Mathematics (M) at University of Texas at Dallas.
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Date Created: 10/12/15
DIFFERENTIALS CHAIN RULE AND DIRECTIONAL DERIVATIVES Gradient If f is a function of two or more variables then the gradient of f denoted as V f is the vector composed of the partial derivatives Example fx y z x3y22 then Vf lt 3x2yzz 2x331 zx3y2 gt In other words V f is the derivative of a function of two or more variables Differentiability A function fx y is differentiable at X0 yo if AZ can be written as AzfXAXfyAysl AX82Ay Where 81 82 gt 0 if AX Ay gt 0 Note V f lt AX Ay gt is called a full differential Theorem Suf cient Condition for Differentiability o If f is a function of X and y and fX and fy are continuous in an open region then f is differentiable at this region Theorem 0 If fx y is differentiable at X0 yo then it is continuous at X0 yo Chain Rule Derivative of Composite Function Derivative of Outer Function Derivative of Inner Function Implicit Differentiation Given z fx y and F X y z 0 then 62 Fx az Fy 5 Fz an 5 F2 Directional Derivatives The directional derivative of fX in the direction of u written as DufX is equal to fXtufX 9 Where f is a function of 11 variables X X1 X2 Xn and u is a unit vector u ltu1 u2 ungt
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