Relation of Rates

Question

If y = x³ + 2x and  , find  when x = 2. Give only the numerical answer. For example, if  = 3, type only 3

Answer 1

Hello Kyle,

If y is a function of x, and x is a function of t, then, by composition of functions, y is also a function of t. To find the derivative of y with respect to t, we use the chain rule in the form

Eq.1) dy/dt = (dy/dx)⋅(dx/dt)

Note that in this equation, if we wish to find dy/dt at a particular value of t (say t_o), we must evaluate dx/dt at t_o, and evaluate dy/dx at the value of x corresonding to t_o (call it x_o). From the information given in the statement of your problem, we wish to find dy/dt at a value of t (not specified) when x = 2 and dx/dt = 5. Given

y = x³ + 2x,

we have

dy/dx = 3x² + 2.

We evaluate this when x = 2 to obtain

dy/dx = 3(2)² + 2

dy/dx = 14

Now substituting the required information into the chain rule (Eq. 1), we have

dy/dt = 14⋅5

dy/dt = 70.

Hope that helps! Let me know if you need any clarification.

William