Differentiation Questions and Answers

A certain function f has inverse function f⁻¹(x) = x³ + 2x + 1. Use this to solve for x in the equation 3f(x) - 1 = 5. (Hint: First solve for what number f(x) must be, then use the inverse to find what number x must be.)

Find the derivative of f(x)=x³ using limits

Suppose f'(x) is continuous over an interval α ≤ x ≤ b, and a < c < d < b. If f'(c) = 5, then (check all that apply) f(x) is decreasing through x = c f(c) = 5 f(c) exists f does not have a minimum at x = c f does not have a maximum at x = c f(x) is increasing through x = c If f'(d) = -5, then (check all that apply) f(d) exists f does not have a maximum at x = d f does not have a minimum at x = d f(d) = -5 f(x) is increasing through x = d f(x) is decreasing through x = d Based on this, we know that for some x between x = c and x = d: f has a maximum f has a minimum f does not exist f'(x) = 0

Use the chain or general power rules to determine the derivative of the following function. f(x)=(5-6x²+2x³)³

Let a≠0 be a complex number, and define the αth branch of zª by f(z) := exp(aLogα(z)). Show that, for z € C \ Rα, we have ƒ'(z) = af(z)/z. This means d/dzzª= aza-1 where we pick the same branch on both sides of the identity.

Find the derivative. f(x) = sinh-¹(-9x) f'(x) =

Use differentials to estimate the amount of paint needed (in m³) to apply a coat of paint 0.04 cm thick to a hemispherical dome with diameter 56 m. (Round your answer to two decimal places.)

If x² + y² + z² = 9, dx/dt= 8, and dy/dt =5, find dz/dt when (x, y, z) = (2, 2, 1).

Use substitution to determine whether the given x-value is a solution of the equation. tan x =√3/3, x= -5π/6 Yes No

Find the derivative, r'(t), of the vector function. r(t) = a + 5tb + t³c

(a) Find the unit vectors that are parallel to the tangent line to the curve y = 8 sin(x) at the point (π/6 ,4). smaller i-component larger i-component (b) Find the unit vectors that are perpendicular to the tangent line. smaller i-component larger i-component

Find f'(x) and find the values(s) of x where f'(x) = 0. f(x) = (2x - 45) (x² +168) f'(x) = x= (Type an integer or a simplified fraction. Use a comma to separate answers as needed.)

An investment of $9,000 earns interest at an annual rate of 8.5% compounded continuously. Complete parts (A) and (B) below. (A) Find the instantaneous rate of change of the amount in the account after 2 year(s). $ (Round to two decimal places as needed). (B) Find the instantaneous rate of change of the amount in the account at the time the amount is equal to $13,500. (Round to two decimal places as needed). $

Find (A) the derivative of F(x)S(x) without using the product rule, and (B) F'(x)S'(x). Note that the answer to part (B) is different from the answer to part (A)- F(x) = x^4 + 1, S(x) = x^5 (A) The derivative of F(x)S(x) is. (B) F'(x)S'(x) =

Find the equation of the line tangent to the graph of f at the indicated value of x. f(x)= 8+ Inx; x = 1

Creamy Bugs Yogurt has found that the cost, in dollars per pound, of the yogurt it produces, is C'(x) = -0.003x + 450, for x ≤ 300, where x is the number of pounds of yogurt produced. Find the total cost of producing 260 pounds of yogurt. A. $116,898.60 B. $449.22 C. $449.61 D. $233,797.20

Use the chain rule to find dz/dt z = xy^9 - x^2y, x = t^2 + 1, y = t^2 - 1

Find f(x+h) - f(x) / h for the function f(x) = 10x - 9

Determine the angle between 0 and 2π that is coterminal to 1020°.

Let (u, v) be the inner product on R² generated by A = [4 1] [-2 1] and let u = (0, -2), v = (4,2). Find (u, v). (u, v) = Current Attempt in Progress Compute the standard inner product on M22 of the given matrices.

Let f(x) = |x|-2 Evaluate the function at the given values. Then select the appropriate graph that represents this function. a. f(-2) = Type your answer here b.f(-1) = Type your answer here c. f(0) = Type your answer here d. f(1) = Type your answer here e. f(2)= Type your answer here f. The graph that represents this function is graph Write your response here.

Consider the following. 3/x - 1/y =7 (a) Find y' by implicit differentiation. y' = (b) Solve the equation explicitly for y and differentiate to get y' in terms of x. y' = (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a). y' =

Find dy/dt for each pair of functions. y=x²-7x, x = t² +3

Find the derivative of the following function. A = 690(1.702)^t

Find an exponential function of the form P(t)= Pon^1/T that models the situation, and then find the equivalent exponential model of the form P(t)=Poe. Doubling time of 4 yr, initial population of 500. Find an exponential function of the form P(t) = Pon^1/T that models the situation. The exponential function is P(t) = 500(2)^1/4 (Use integers or fractions for any numbers in the expression.) Find the equivalent exponential model of the form P(t) = Poe^rt.

Differentiate the function. f(x) = x²(x + 4) f'(x) = 4x³ + 12x²

Find dy/dx by implicit differentiation. x² - 14xy + y² = 14

Find the derivative of the function. f(t)- 7t sin(ut)

Let f(u) = u³ and g(x)=u=6x^5 +5. Find (fog)'(1). (fog)'(1) = (Type an exact answer.)

Find dy/dt for each pair of functions. y=x²-2x, x= t² +4 dy/dt=

The demand for a new computer game can be modeled by p(x) = 41-8 In x, for 0 ≤x≤ 800, where p(x) is the price consumers will pay, in dollars, and x is the number of games sold, in thousands. Recall that total revenue is given by R(x)=x p(x). Complete parts (a) through (c) below.

Use the chain rule or general power rule to determine the derivative of the following f(x)=(2x³-6x²)^ 3 2.) y=√6x² +8

For the polynomial below, -1 is a zero. f(x)=x²³ - 5x² + 6 Express f(x) as a product of linear factors.

Find all ercepts and y-intercepts of the graph of the function. f(x) = 2x² +9x+10 If there is more than one answer, separate them with commas. Click on "None" if applicable. x-intercept(s): 0 y-intercept(s): 0 None 0/0 Xx 0/6 0,0.... S 0/6 ?

- 23x² + 48 x²6 Determine the intervals on which f is decreasing. Consider f(x) Of is decreasing on: Of is decreasing nowhere. Determine the intervals on which f is increasing. Of is increasing on: Of is increasing nowhere. Determine the value and location of any local minimum of f. Enter the solution in (x, f(x)) form. If multiple solutions exist, use a comma-separated list to enter the solutions. of has a local minimum at: Of has no local minimum. Determine the value and location of any local maximum of f. Enter the solution in (x, f(x)) form. If multiple solutions exist, use a comma-separated list to enter the solutions. of has a local maximum at: Of has no local maximum.

Consider h(y) = -7 cos y +7√3 siny on [0, 2π]. Determine the intervals on which h is decreasing. Oh is decreasing on: Oh is decreasing nowhere. Determine the intervals on which h is increasing. Oh is increasing on: h is increasing nowhere. Determine the value and location of any local minimum of f. Enter the solution in (y, h(y)) form. If multiple solutions exist, use a comma-separated list to enter the solutions. Oh has a local minimum at: Oh has no local minimum. Determine the value and location of any local maximum of f. Enter the solution in (y, h(y)) form. If multiple solutions exist, use a comma-separated list to enter the solutions. Oh has a local maximum at: h has no local maximum.

Find the derivative of the function. f(x) = sin( 6

Find all critical numbers of the function g(x)=x²-14x². O critical numbers: x=0, x=√14, x= -√14 O critical numbers: x=0, x=√√7, x= -√7 O critical numbers: x = √14, x = O critical numbers: x = no critical numbers = -√14 √7,x= -√7

Find the relative extremum of f(x)=-3x² +30x+8 by applying the First Derivative Test.

Complete two iterations of Newton's Method for the function using the given initial guess. f(x) = x² - 4₁ x₁ = 1.1

Find the marginal revenue function. R(x) = 3x -0.04x² R'(x) =

Find the derivative of the function. f(θ) = cos(θ²) f'(θ) =

Find y" by implicit differentiation. Simplify where possible. x² + 3y² =3 y" =

Find dy/dx by implicit differentiation. tan(x - y) =y/2+x²

1. Which of the following statement is false? (a) d/dx In(2x) = d/dx In(x) for x > 0. (b) d/dx csc(x) = cot(x) csc (x) for x = (0, π). (c) d/dx sin(x)/ cos(x) - = 1 + tan² (x) for x = (-π /2, π/2). (d) d/dx2* = ln(2) - 2^x for x € (-∞, ∞). (e) d/dx (x³ + 2x + 1) = 3x² + 2 for x € (-∞0,00).

Find the equation of the tangent line to f(x) = 1 + e X + X 2 at x = 0.

let f(x) = sin x. Write an expression for f(x) using the definition of the derivative, but don't evaluate the limit.

Find the total change of f(x) between x = 1 and x = 4, when df/dx =1/x² 1.25 0.75 0.25 -0.75

If the number of fruit flies F increases at a rate that is proportional to the number of flies present, write the differential equation that describes the rate of change in the number of fruit flies at a given time, t. Do not solve. dF/dt = k/F dF/dt = -k/F dF/dt = kF dF/dt = tF

The interest on an account increases exponentially. If an initial investment of $5000 earns $900 in two years, what is the rate of increase at the end of that second year? A. $500.00 B. $488.52 C. $462.17 D. $450.00