For a positive constant a find dxdy, where y=ai+t1, and x=(t+t1)a $
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Step-by-Step Solution
Step 1: Differentiate y with respect to t
We need to find dxdy. Since both y and x are functions of t, we can use the chain rule: dxdy=dtdy⋅dxdt. First, let's differentiate y with respect to t. We use the formula for differentiating bf(t), which is bf(t)lnb⋅f′(t). Here, b=a and f(t)=t+t1. The derivative of t+t1 is 1−t21.
Step 2: Differentiate x with respect to t
Next, we differentiate x with respect to t. We use the chain rule for differentiating ua, where u=t+t1. The derivative is aua−1⋅dtdu. So, we get a(t+t1)a−1 multiplied by the derivative of t+t1, which is 1−t21.
Step 3: Find dt/dx
Since we need dxdt for the chain rule, we take the reciprocal of dtdx.
Step 4: Apply the Chain Rule
Now we multiply dtdy by dxdt to find dxdy. We substitute the expressions we found in the previous steps.
Step 5: Simplify the expression
We can cancel out the common term (1−t21) from the numerator and the denominator, assuming t=±1. This leaves us with the simplified expression for dxdy.