William P. answered • 01/02/20

University Math Instructor and Experienced Calculus Tutor

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 = x^{3} + 2x,

we have

dy/dx = 3x^{2} + 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