Question
Find \( \frac{d y}{d x} \) if \( x=a \sin ^{2} t \) and \( y=a \cos ^{2} t, 9 \) is a constant
Ask by Pritchard Little. in Nigeria
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\[
\frac{d y}{d x} = -1
\]
Solution
To find \( \frac{d y}{d x} \), we can use the chain rule. We will first find \( \frac{d y}{d t} \) and \( \frac{d x}{d t} \), and then use the relationship:
\[
\frac{d y}{d x} = \frac{d y/d t}{d x/d t}
\]
### Step 1: Find \( \frac{d y}{d t} \)
Given \( y = a \cos^2 t \), we can differentiate it with respect to \( t \):
\[
\frac{d y}{d t} = a \cdot 2 \cos t \cdot (-\sin t) = -2a \cos t \sin t
\]
### Step 2: Find \( \frac{d x}{d t} \)
Given \( x = a \sin^2 t \), we can differentiate it with respect to \( t \):
\[
\frac{d x}{d t} = a \cdot 2 \sin t \cdot \cos t = 2a \sin t \cos t
\]
### Step 3: Find \( \frac{d y}{d x} \)
Now we can substitute \( \frac{d y}{d t} \) and \( \frac{d x}{d t} \) into the formula for \( \frac{d y}{d x} \):
\[
\frac{d y}{d x} = \frac{-2a \cos t \sin t}{2a \sin t \cos t}
\]
### Step 4: Simplify
The \( 2a \sin t \cos t \) terms cancel out:
\[
\frac{d y}{d x} = -1
\]
Thus, the final result is:
\[
\frac{d y}{d x} = -1
\]
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Simplify this solution Beyond the Answer
To find \( \frac{dy}{dx} \), we can use the chain rule and parametric differentiation. First, we differentiate \( x \) and \( y \) with respect to \( t \): \[ \frac{dx}{dt} = \frac{d}{dt}(a \sin^2 t) = a \cdot 2 \sin t \cos t = a \sin(2t) \] \[ \frac{dy}{dt} = \frac{d}{dt}(a \cos^2 t) = a \cdot 2 \cos t (-\sin t) = -2a \sin t \cos t = -a \sin(2t) \] Now we can find \( \frac{dy}{dx} \) using the chain rule: \[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{-a \sin(2t)}{a \sin(2t)} = -1 \] Thus, the derivative \( \frac{dy}{dx} \) is: \[ \frac{dy}{dx} = -1 \]
