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- f(x,y,z)=5sin(xyz), (a,b,c)=(1,1,π/4), and v⃗ =(3,-√2,3). Calculate the directional derivative of f at the point (a,b,c)=(1,1,π/4) in the direction defined by v⃗. Find the direction at (a,b,c)=(1,1,π/4) in which the rate of change of f is greatest. Find the maximum rate of change. Fill in the blank: f decreases the most at (a,b,c)=(1,1,π/4) in the direction off(x,y)=3 e^x cos y, (a,b)=(0,π/4), and v⃗ =(2,3). Calculate the directional derivative of f at the point (a,b) in the direction defined by v⃗ . Find the direction at (a,b) in which the rate of change of f is greatest. Find the maximum rate of change. Fill in the blank: f decreases the most at (a,b) in the direction of(a) Applications of partial derivative: ii) A bee was flying upward along the curve that is the section ofz = x^4 + xy^3 + 12 with the plane x = 1. At the point (1, -2, 5), it went off on the tangent line. Where did the bee hit the xz-plane
- f(x,y)=e^xcosy, (a,b)=(0,π/4), and v⃗ =(2,3). Calculate the directional derivative of f at the point (a,b) in the direction defined by v⃗ . Find the direction at (a,b) in which the rate of change of f is greatest. Find the maximum rate of change. Fill in the blank: f decreases the most at (a,b) in the direction ofZero directional derivative In what directions is the derivative of ƒ(x, y) = (x2 - y2) / (x2 + y2) at P(1, 1) equal to zero?f(x,y,z)=x.y.zxy for the function fzx(x,y,z) calculate derivative.
- a) find the rate of change in f(x,y) when moving in the direction of the vector (1,3) b) find the directional derivative of f(x,y) when moving in the direction of maximum increase c) find the direction of maximum decrease in f(x,y)Derivative rules Suppose u and v are differentiable functions at t = 0 with u(0) = ⟨0, 1, 1⟩, u'(0) = ⟨0, 7, 1⟩, v(0) = ⟨0, 1, 1⟩, and v'(0) = ⟨1, 1, 2⟩ . Evaluate the following expression.Differentiate Derivative ofouter function Derivative ofinner function Differentiation 1. y = sin (2x) cos (2x) 2 2 cos (2x) 2. y = (4t2 – 3t + 2) -2 3. f(x) = (5x4 – 7) 2/3 4. y = (sin 3x)2 Directions: Complete the table by determining the derivatives of the outerand inner functions of the given functions and finding the differentiationusing the Chain Rule. Answer numbers 2, 3, and 4 please
- Using Cauchy-Riemann equations, show that the function f(z)=(z+6)^2 is differentiable everywhere.[Directional Derivatives] Calculate the following directional derivatives. w(x, y, z) = xy^2 / z^2 at the point (6, 1, 2) in the direction toward (5, 2, 5).Chain Rule with one independent variable Use Theorem 15.7 to find the following derivatives. dz/dt, where z = x sin y, x = t2, and y = 4t3