Let a E R. Let f be a function. Assume f is differentiable at a. 1 Assume f is never 0. Let g be the function defined by the equation g(x)% = f(x) %3D -f (a) f(a)? Write a proof directly from the definition of derivative, without using any of the differentiation Prove that g is differentiable at a and that g'(a) = %3D

Consider the graph of f(x), the derivative of f(x). Find each of the following: You must use interval notation as needed. Use a comma to separate answers as needed. If a question has no solution, type none for the answer. y = f'(x) 4 3 2 1. f is increasing in the interval(s) 2. f is decreasing in the interval(s) -5 -4 -3 -2 -11 -2 3. f has a relative maximum at x = 5Ay 4. f is concave up in the interval(s): 11 6. f has inflection points at x = 1 -4 -5- 5. f is concave down in the interval(s): 1 2 3 4 10

Explain the differences between computing the derivatives of functions that are defined implicitly and explicitly. Select each of the following that correctly describes the differences O A. When computing the derivative of an explicitly defined function y = f(x), the result dy / dx depends only on x. When computing the derivative of an implicitly defined function, the result dy / dx depends only on y. O B. When computing the derivative of an explicitly defined function y = f(x), the result dy / dx depends only on x. When computing the derivative of an implicitly defined function, the result dy / dx may depend on both x and y. C. To compute the derivative of an explicitly defined function y = f(x), use the rules of differentiation to differentiate y with respect to x. To compute the derivative of a function defined implicitly by an equation, write the independent variable y as a function of the dependent variable and x, use the chain rule to differentiate each term of the equation with…

3) Find the derivative of the functions. Make sure to show all work, and explain which differentiation rules you used and why: a. f(x) = xe* %3D sin x b. g(x) = x4+1 c. h(x) = (2x² + 3x – 1)(6+x – 2x³) x³+3x-2 d. p(x) : e. r(x) = x³ tan x f. s(x) = 3 sec x – 5 csc x cOS X в т(x) — ex