Partial Derivatives (§14.3)Tangent Planes and Differentiability (§14.4)Chain Rule (§14.5)Partial Derivatives (§14.3)Tangent Planes and Differentiability (§14.4)Chain Rule (§14.5)Math 241, Spring 2014Jayadev S. AthreyaSpring 2014Jayadev S. Athreya Math 241, Spring 2014Partial Derivatives (§14.3)Tangent Planes and Differentiability (§14.4)Chain Rule (§14.5)How functions changeWe will be interested in how functions change, and how toapproximate them by functions we can understand. In1-variable, the derivative played two roles:Rate of change f0(x) tells us how fast the function f waschanging at x.Linear approximation f0(x) is the slope of the tangent line at(x, f (x)), which gives us a good approximation tothe function.Jayadev S. Athreya Math 241, Spring 2014Partial Derivatives (§14.3)Tangent Planes and Differentiability (§14.4)Chain Rule (§14.5)Partial derivativesGiven a function of 2 variables f(x, y ), we can hold one of thevariables fixed and let the other one change, to get functions ofone-variable, and can compute derivatives of those functions:∂∂xThe partial derivative with respect to x at (a, b),denoted∂f∂x(a, b) is given by (if the limit exists)∂f∂x(a, b) = limh→0f (a + h, b) −f (a, b)hOther notation Sometimes we write (Dxf )(a, b), or fx(a, b)Jayadev S. Athreya Math 241, Spring 2014Partial Derivatives (§14.3)Tangent Planes and Differentiability (§14.4)Chain Rule (§14.5)Partial derivativesGiven a function of 2 variables f(x, y ), we can hold one of thevariables fixed and let the other one change, to get functions ofone-variable, and can compute derivatives of those functions:∂∂yThe partial derivative with respect to y at (a, b),denoted∂f∂h(a, b) is given by (if the limit exists)∂f∂y(a, b) = limh→0f (a, b + h) −f(a, b)hOther notation Sometimes we write (Dyf )(a, b), or fy(a, b)Jayadev S. Athreya Math 241, Spring 2014Partial Derivatives (§14.3)Tangent Planes and Differentiability (§14.4)Chain Rule (§14.5)ExampleJayadev S. Athreya Math 241, Spring 2014Partial Derivatives (§14.3)Tangent Planes and Differentiability (§14.4)Chain Rule (§14.5)More Examplesf (x, y) = ex+yf (x, y) = sin(xy)f (x, y) = ex2f (x, y) = x3y2Jayadev S. Athreya Math 241, Spring 2014Partial Derivatives (§14.3)Tangent Planes and Differentiability (§14.4)Chain Rule (§14.5)Slopes of Tangent Lines Jayadev S. Athreya Math 241, Spring 2014Partial Derivatives (§14.3)Tangent Planes and Differentiability (§14.4)Chain Rule (§14.5)Partial derivatives: 3DGiven a function of 3 variables f (x, y, z), we can hold all but one ofthe variables fixed and let the other one change, to get functions ofone-variable, and can compute derivatives of those functions:∂∂xThe partial derivative with respect to x at (a, b, c),denoted∂f∂x(a, b, c) is given by (if the limit exists)∂f∂x(a, b, c) = limh→0f (a + h, b, c) − f (a, b, c)hOther notation Sometimes we write (Dxf )(a, b, c), or fx(a, b, c)Jayadev S. Athreya Math 241, Spring 2014Partial Derivatives (§14.3)Tangent Planes and Differentiability (§14.4)Chain Rule (§14.5)Partial derivatives: 3DGiven a function of 3 variables f (x, y, z), we can hold all but one ofthe variables fixed and let the other one change, to get functions ofone-variable, and can compute derivatives of those functions:∂∂yThe partial derivative with respect to y at (a, b),denoted∂f∂y(a, b) is given by (if the limit exists)∂f∂y(a, b, c) = limh→0f (a, b + h, c) −f (a, b, c)hOther notation Sometimes we write (Dyf )(a, b, c), or fx(a, b, c)Jayadev S. Athreya Math 241, Spring 2014Partial Derivatives (§14.3)Tangent Planes and Differentiability (§14.4)Chain Rule (§14.5)Partial derivatives: 3DGiven a function of 3 variables f (x, y), we can hold all but one of thevariables fixed and let the other one change, to get functions ofone-variable, and can compute derivatives of those functions:∂∂zThe partial derivative with respect to y at (a, b),denoted∂f∂z(a, b) is given by (if the limit exists)∂f∂z(a, b, c) = limh→0f (a, b, c + h) − f (a, b, c)hOther notation Sometimes we write (Dzf )(a, b, c), or fz(a, b, c)Jayadev S. Athreya Math 241, Spring 2014Partial Derivatives (§14.3)Tangent Planes and Differentiability (§14.4)Chain Rule (§14.5)Examplesf (x, y, z) = z2x + zy + xyf (x, y, z) = zex2+yf (x, y, z) = sin(xz) + cos(xy)f (x, y, z) = ex2+y2f (x, y, z) = x3y2+ cos zJayadev S. Athreya Math 241, Spring 2014Partial Derivatives (§14.3)Tangent Planes and Differentiability (§14.4)Chain Rule (§14.5)Partial derivatives: n-dimensionsGiven a function of n variables f (x1, . . . , xn), we can hold one of thevariables fixed and let the other one change, to get functions ofone-variable, and can compute derivatives of those functions:∂∂xiThe partial derivative with respect to xiatp = (p1, . . . , pn), denoted∂f∂x(p) is given by (if the limitexists)∂f∂xi(p) = limh→0f (p1, . . . , pi+ h, . . . , pn) −f (p)hOther notation (Dxif )(p) or (Dif )(p), or fxi(p)Jayadev S. Athreya Math 241, Spring 2014Partial Derivatives (§14.3)Tangent Planes and Differentiability (§14.4)Chain Rule (§14.5)Tangent PlanesConsider
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