Differentiating the function f(x), using the Chain Rule. f(x) = {Vx5 + 6x f'(x) = =(x5 + 6x) 3 A 5x4 x + 6x) 3 B f'(x) = 5x4 + 6 C f'(x) = 37 (x5 + 6x)² (5x4 + 6) D f'(x) = 2/ x + 6x 2/3
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A: Topic : critical number
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- 1. How do you find the linearization of f(x) at x = a? Illustrate with a concrete exampleHow do you remember the methods for partial derivatives in the chain rule for several variables?Find the linearization of f(x)=ln(x2-3) at suitably chosen integer near X=2.1. Then use the linearization to estimate the value of f(2.1).
- A first order nonlinear system is described by the equationẋ = −f(x)where f(x) is a continuous and differentiable nonlinear function thatsatisfies the following:f(0) = 0;f(x) > 0 for x > 0;f(x) < 0 for x < 0.Use the Lyapunov function V(x) = x2/2 to show that the system isstable near the origin.If F admits continuous partial derivatives and the equation F ( 3x - y, z2 - x2)= 0 defines z implicitly as a function of x and y, then taking u = 3x - y and w = z2 - 2x then the value of the expression image1 corresponds to image 2Explain how successive over relation is obtained from the Gauss-Seidel method.
- Create your rational function such that you can use both log rule and partial fraction decomposition to integrate it. And use these two methods to integrate it. (Please by hand). step-by stepSole the nonhomogeneous linear system using the Laplace Transform (or any method you like) using the given initial conditions.Use algebraic methods to determine the critical value(s) of f(x)= x/(x^2-x-2). Give your answers in exact form.