Problem 3 - Essential Calculus

Question

Find \(d y / d x\). $$ x=\sin ^{2} \theta, y=\cos ^{2} \theta $$

Step-by-Step Solution

Answer

The derivative \(d y / d x\) is \(-1\).

Step 1: Differentiate \(x=\sin^2\theta\) and \(y=\cos^2\theta\) with respect to \(\theta\)

This will be done using the chain rule.The derivative of \(x=\sin^2\theta\) with respect to \(\theta\) is \(d x / d \theta = 2\sin(\theta) \cos(\theta)\), by differentiating the outer function first then multiplying by the derivative of the inner function.The derivative of \(y=\cos^2\theta\) with respect to \(\theta\) is \(d y / d \theta = -2\cos(\theta) \sin(\theta)\)

Step 2: Find \(d y / d x\)

The derivative \(d y / d x\) can be found using the relation \((d y / d x)=(d y / d \theta) / (d x / d \theta)\). Using the results from Step 1, we get \(d y / d x = -2\cos(\theta) \sin(\theta) / 2\sin(\theta) \cos(\theta)\)

Step 3: Simplify

Simplify the expression obtained in Step 2 to give your final result. Since the numerator and denominator are identical, their quotient is simply \(-1\).