chain rule/total derivative?

Question

I have to use the chain rule to obtain the total derivative w.r.t. t

a) f(x,y) = x + y^2 ...when x = t^2 and y = t^3

b) f(x,y) = xlny + ylnx ...when x = t + 1 and y = lnt

any help on this is appreciated.

Answer 1

Using the definition of total derivative, df/dt = (∂f/∂x)(∂x/∂t) + (∂f/∂y)(∂y/∂t).

a) Given the expressions for x and y, (∂x/∂t) = 2t and (∂y/∂t) = 3t^2. In addition, (∂f/∂x) = 1 and (∂f/∂y) = 2y. Note that the partial differentiation operator treats variables other than the one being differentiated with respect to as constants. So df/dt = (∂f/∂x)(∂x/∂t) + (∂f/∂y)(∂y/∂t) = 1*2t + 2y*3t^2 = 2t + 2(t^3)*3t^2 = 2t + 6t^5.

Of course, you can also substitute as your first step, giving you f(t) = t^2 + (t^3)^2 = t^2 + t^6, from which you can easily get df/dt = 2t + 6t^5, the same result.

b) would be done in the same way, using either method. (But if you're expected to use the chain rule, you should use the first method.) Note that you'll also need to use the product rule.