Differentiation Questions and Answers

Let g(z)=5ze^3z Determine the critical values of g. If multiple such values exist, enter the solutions using a comma-separated list.

Let h(y) = 10y - 5 Determine the critical values of h. If multiple such values exist, enter the solutions using a comma-separated list.

Let h(w) = -3.8ʷw⁸ Determine the critical values of h. If multiple such values exist, enter the solutions using a comma-separated list.

Let g(v) = -6v² + 18v³ +7 - 16v Determine the critical values of g. If multiple such values exist, enter the solutions using a comma-separated list.

Let f(u) = -14u(u + 10)1/3 Determine the critical values of f. If multiple such values exist, enter the solutions using a comma-separated list. u = No critical values exist.

Let g(x)=x√9+10x Determine the critical values of g. If multiple such values exist, enter the solutions using a comma-separated list. x = No critical values exist.

Find dy/dx by implicit differentiation. x^2/ x+y= y² + 8

Find the derivative of the function. f(x) = (2x - 6)^4(x² + x + 1)^5

Find the derivative of the function. f(t) = 5t sin(nt) f'(t) =

Finddy/dx by implicit differentiation. x sin(y) + y sin(x) = 1 dy/dx=

Find dy/dx by implicit differentiation. x²-18xy + y² = 18

Find dy/dx by implicit differentiation. xe^y = x-y dy/dx=

Find the derivative of the function. y = e^tan(θ) y' =

Find the derivative of the function. F(t) = e^8tsin(2t) F'(t)=

Find the derivative of the function. f(z) = e^z/(z - 9)

Find dy/dt using the given values. y = x³ - 4x for x = 5, dx/dt = 3. dy/dt=

Find the derivative of the function. F(x) = (9x^6 + 8x³)^4

Find the derivative of the function. f(x) = √3x + 4 f'(x) =

Find the derivative of the function. h(t) = (t + 3)^2/3 (2t² - 1)³ h'(t) =

Use the implicit differentiation to find the derivative dy/dx= y', given the implicit equations:e^3xy = 3x + y²

Find dy/dx by implicit differentiation. y cos(x) = 4x² + 5y² y' =

Find f. f'(t)= sec(t)(sec(t) + tan(t)), - π/2 < t < π/2, f(π/4)=-1

For the given composite function, identify the inner function, u = g(x), and the outer function, y = f(u). (Use non-identity functions for f(u) and g(x).) y = tan(πx) (f(u), g(x)) = Find the derivative, dy/dx

Two trains leave from the same station, one going north at 30 miles/hr and one going west at 40 miles/hr. The one going west leaves 1 hour later than the one going north. When the northbound train is 120 miles from the station, how fast is the distance between the trains changing?

Find the first and second derivative of the function. G(r) = √r + √r

If R4) 13 and f'(x) z 2 for 4 ≤ x ≤ 6, how small can f(6) possibly be?

The function p(t) = 2000t/ 4t+75 gives the population p of deer in an area after t months. a) Find p'(5), p'(25), and p'(50). b) Find p''(5), p''(25), and p''(50). c) Interpret the meaning of your answers to part (a) and (b). What is happening to the population of deer in the long term?

The equation of motion of a particle is s= t³ - 27t, where s is measured in meters and t is in seconds. (a) Find the velocity and acceleration as functions of t. v(t) = a(t) = (b) Find the acceleration, in m/s2, after 9 seconds. m/s² (c) Find the acceleration, in m/s2, when the velocity is 0. m/s²

Consider the following. Find f'(x). f'(x) = f(x) = x5 - 4x³ + x-1 Compare the graphs of f and f' and use them to explain why your answer is reasonable f'(x) = 0 when f_________ f'(x) is positive when f ____ f'(x) is negative when f ______

Differentiate. f(x)= = x/( x + c/x) f'(x) =_______

Let f(x) = (1 + 3x²)(x - x²). Find the derivative by using the Product Rule. f'(x) = Find the derivative by multiplying first. f'(x) = Do your answers agree? Yes No

Differentiate. h(θ) = θ² sin(θ) h'(θ)=

Let the function f be defined by f(x)=tan(x) -1/sec(x) (a) Use the Quotient Rule to differentiate the function F'(x). f'(x) = (b) Simplify the expression for f(x) by writing it in terms of sin(x) and cos(x), and then find f'(x). F'(x) = (c) Show that your answers to parts (a) and (b) are equivalent? Yes No

Differentiate. y = sec(θ) tan(θ) y'=

Write the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).] (Use non-identity functions for f(u) and g(x).) y=e³√x (f(u), g(x)) = Find the derivative dy/dx=

Graph the parabola. y=-2x²-12x-14 Plot five points on the parabola: the vertex, two points to the left of the vertex, and two points to the right of the vertex. Then click on the graph-a-function button.

Find f '(3), where f(t) = u(t) · v(t), u(3) = (1, 2, -2), u'(3) = (7, 1, 4), and v(t) = (t, t², t³). f'(3) =

Finddy y =30 +10x²-x³ dy=____(Simplify your answer.)

Consider the following vector equation. r(t) = 4 sin(t)i - 3 cos(t)j (a) Find r'(t). r' (t) =

Find the derivative, r'(t), of the vector function. r(t) = (e^-t, 6t - t³, In(t)) r'(t) =

Use the product or the quotient rule to determine the derivative of the following function. f(x)=(x³-3x²) (2x³+4x²)

Find the average cost function if cost and revenue are given by C(x) = 112+ 4.2x and R(x) = 9x-0.04x². The average cost function is C(x) =______

Find the slope and an equation of the line tangent to the graph of the given function at the indicated f(x)=(√x²)^1/3+5/3 x ,(8,4)

A non-linear system is governed by the following differential equation dx +x²u; x(0) 0.1; u = 1 dt a. (5) Solve the diff-eq numerically and plot the response b. (5) Linearize the diff-eq about u=1. Write down the diff-eq c. (5) Plot the numerical solution of linearized differential equation. d. (5) Solve the diff-eq analytically (variable separable) for u=1.

Find the indicated higher - ordered derivative. Find: f'''(x), where f(x) = x5 - 2x4+ 3x³ - 4x² + 5

Use the basic rules for differentiation to obtain the derivative of the following function. f(x) =3x6 - 4x4+ x - 12

The ideal gas law for a certain amount of gas relates the pressure P (in N/m2) to the temperature T (in Kelvin K) and the volume V (in m³) of a container via P= T/V. Suppose the container has an initial volume of 1/5 m³ and an initial temperature of 300 degrees Kelvin. If the volume increases by 0.04 m³ and the temperature increases by 7 degrees, by how much does the pressure change?