SOLUTION: if y = t ^2 + x ^2 - 2 xt ^3 , then (d ^2 y)/(dx . dt) = .....

Question 1208656: if y = t ^2 + x ^2 - 2 xt ^3 , then (d ^2 y)/(dx . dt) = .....
Answer by Shin123(626) (Show Source): You can put this solution on YOUR website!
Note that (d^2y)/(dxdt)=(d/dx)(dy/dt). This means we take the partial derivative with respect to t first, and then with respect to x.
First, we calculate dy/dt. Let's do one term at a time. The derivative of t^2 with respect to t is 2t by the power rule. The derivative of x^2 with respect to t is 0, since x is a constant with respect to t. The derivative of -2xt^3 with respect to t is -6xt^2, since we use the power rule again, and we again treat x as a constant. Adding up all of this gives -6xt^2+2t.
Then, we take the partial derivative of that with respect to x. The derivative of -6xt^2 with respect to x is -6t^2, since all the t terms are considered constant with respect to x. The derivative of 2t with respect to x is 0, since t is constant with respect to x. Therefore, our final answer is -6t^2.
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