2.5 Q-9

Question Statement

We are given the following equations:

$$x = a(\cos t + \sin t)$$ $$y = a(\sin t - t \cos t)$$

We need to find:

$$\frac{d y}{d x}$$

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Background and Explanation

In this problem, we are asked to differentiate two expressions with respect to $t$ and then find $\frac{d y}{d x}$, which requires applying the chain rule.

• The chain rule states that $\frac{d y}{d x} = \frac{d y}{d t} \times \frac{d t}{d x}$.
• We’ll differentiate $x$ and $y$ with respect to $t$ and then apply the formula for $\frac{d y}{d x}$.

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Solution

Step 1: Differentiate $x = a(\cos t + \sin t)$ with respect to $t$

We are given:

$$x = a(\cos t + \sin t)$$

Differentiating with respect to $t$:

$$\frac{d x}{d t} = a(-\sin t + \cos t)$$

Thus, the derivative of $x$ with respect to $t$ is:

$$\frac{d x}{d t} = a(-\sin t + \cos t)$$

Step 2: Differentiate $y = a(\sin t - t \cos t)$ with respect to $t$

Next, we differentiate the expression for $y$:

$$y = a(\sin t - t \cos t)$$

Differentiating with respect to $t$:

$$\frac{d y}{d t} = a[\cos t - (\cos t - t \sin t)]$$

Simplifying:

$$\frac{d y}{d t} = a[\cos t - \cos t + t \sin t]$$ $$\frac{d y}{d t} = a t \sin t$$

Step 3: Apply the chain rule to find $\frac{d y}{d x}$

Now we can apply the chain rule:

$$\frac{d y}{d x} = \frac{d y}{d t} \times \frac{d t}{d x}$$

We already know:

$$\frac{d y}{d t} = a t \sin t$$ $$\frac{d x}{d t} = a(-\sin t + \cos t)$$

Thus, $\frac{d t}{d x}$ is the reciprocal of $\frac{d x}{d t}$:

$$\frac{d t}{d x} = \frac{1}{a(\cos t - \sin t)}$$

Now, substitute the values into the chain rule formula:

$$\frac{d y}{d x} = a t \sin t \times \frac{1}{a(\cos t - \sin t)}$$

Simplifying:

$$\frac{d y}{d x} = \frac{t \sin t}{\cos t - \sin t}$$

Step 4: Final Expression

Thus, we have:

$$\frac{d y}{d x} = \frac{t \sin t}{\cos t - \sin t}$$

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Key Formulas or Methods Used

• Chain Rule: Used to relate the derivatives of $x$ and $y$ with respect to $t$ and then express $\frac{d y}{d x}$.

$$\frac{d y}{d x} = \frac{d y}{d t} \times \frac{d t}{d x}$$

• Differentiation:
  • $\frac{d}{d t}(\sin t) = \cos t$
  • $\frac{d}{d t}(t \cos t) = \cos t - t \sin t$

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Summary of Steps

1. Differentiate $x = a(\cos t + \sin t)$ to find $\frac{d x}{d t}$.
2. Differentiate $y = a(\sin t - t \cos t)$ to find $\frac{d y}{d t}$.
3. Apply the chain rule to find $\frac{d y}{d x}$.
4. Simplify the expression for $\frac{d y}{d x}$ to obtain the final result.

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