If x=t2 and y=t3, then dx2d2y is equal to :
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Step-by-Step Solution
Step 1: Find the first derivative dxdy
We are given x and y in terms of a parameter t. To find dxdy, we use the chain rule for parametric equations. First, we find the derivatives of x and y with respect to t.
Step 2: Calculate dtdx and dtdy
Differentiating x=t2 with respect to t gives 2t. Differentiating y=t3 with respect to t gives 3t2.
Step 3: Substitute to find dxdy
Now we substitute the derivatives dtdx and dtdy into the chain rule formula to find dxdy. We simplify the expression by canceling out t.
Step 4: Find the second derivative dx2d2y
To find the second derivative dx2d2y, we need to differentiate dxdy with respect to x. Since dxdy is a function of t, we again use the chain rule for parametric differentiation.
Step 5: Calculate dtd(dxdy) and substitute
First, we differentiate dxdy=23t with respect to t, which gives 23. Then, we divide this result by dtdx=2t to obtain the second derivative dx2d2y.