Question

Find df using the chain rule and direct substitution. dt f(x, y) V x2 + y2, y = t2, x=t df chain rule dt direct substitution df dt

Answer 1

To find df using the chain rule and direct substitution, differentiate x^2 + y^2 with respect to x and y, and then substitute y = t^2. The resulting expression is 2x + 4t^3.

Step-by-step explanation:

To find df using the chain rule and direct substitution, we first find the partial derivative of f with respect to x, and the partial derivative of f with respect to y. Using the chain rule, we differentiate x^2 + y^2 with respect to x and y, and then substitute y = t^2. This gives us:

df/dt = (df/dx)(dx/dt) + (df/dy)(dy/dt)

df/dt = 2x(1) + 2y(2t)

df/dt = 2x + 4yt

Substituting y = t^2 into the expression, we get:

df/dt = 2x + 4t^3