Find dxdy, if x=at2,y=2at.
Get the complete, step-by-step math solution for: "Find (d y)/(d x), if x=a t^{2}, y=2 a t.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Differentiate x with respect to t
We are given x as a function of t, specifically x=at2. To find dtdx, we differentiate at2 with respect to t. Using the power rule for differentiation, dtd(tn)=ntn−1, we get 2at.
Step 2: Differentiate y with respect to t
Next, we differentiate y with respect to t. Given y=2at, its derivative with respect to t is 2a, as a is a constant and the derivative of t with respect to t is 1.
Step 3: Apply the Chain Rule
To find dxdy, we use the chain rule for parametric equations. This rule states that dxdy can be found by dividing dtdy by dtdx.
Step 4: Substitute and Simplify
Now, we substitute the expressions we found for dtdy and dtdx into the chain rule formula. After substitution, we simplify the expression by canceling out common terms.