4. Applications of Derivatives

Given $x=e^t$ and $y=t{e}^{-t}$, which of the following is the correct value of $\frac{dy}{dx}$ at $t=1$?

If x = y³ – y and dy/dt = 5, then what is dx/dt when y = 2?

Assume that y = 5x and dx/dt = 2. Find dy/dt

Which of the following gives the equations of both lines through the point $\left(2,,,,-,3\right)$ that are tangent to the parabola $y=x^2+x$?

Given the equation $x+3y{y}^{13}$=$y$, what is $\frac{dy}{dx}$ at the point $(2,8)$?

If x²y³ = 4/27 and dy/dt = ¹/₂, then what is dx/dt when x = 2?

A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. (Hint: Surface area=4πr².)

Explain the difference between the average rate of change and the instantaneous rate of change of a function f.

Given the equation $e^x-y=x{y}^3+e^2-18$, what is the value of $\frac{dy}{dx}$ at the point $(2,2)$?

A sphere is growing at a rate of $50\frac{\operatorname{\mathrm{cm}}^3}{s}$. At what rate is the radius of the sphere increasing when the radius is $5\operatorname{\mathrm{cm}}$?