Math / Calculus

Question7. Find $\frac{d y}{d x}$ if $y^{2}-3 x y+x^{2}=7$ . (a) $\frac{2 x+y}{3 x-2 y}$ (b) $\frac{3 y-2 x}{2 y-3 x}$ (c) $\frac{2 x}{3-2 y}$ (d) $\frac{2 x}{y}$ (e) None of these

Studdy Solution

STEP 1

What is this asking? We need to find the derivative of $y$ with respect to $x$ , written as $\frac{dy}{dx}$ , given the equation $y^2 - 3xy + x^2 = 7$ . Watch out! Remember that $y$ is a function of $x$ , so we'll need to use the chain rule when differentiating terms involving $y$ .
Don't forget to treat $y$ as $y(x)$ !

STEP 2

1. Implicit Differentiation
2. Isolate and Simplify

STEP 3

Let's differentiate both sides of the equation $y^2 - 3xy + x^2 = 7$ with respect to $x$ .
Remember, we're thinking of $y$ as a function of $x$ , so we'll need the chain rule for terms involving $y$ .

STEP 4

$\frac{d}{dx}(y^2) - \frac{d}{dx}(3xy) + \frac{d}{dx}(x^2) = \frac{d}{dx}(7)$ $2y \cdot \frac{dy}{dx} - \left( 3y + 3x \cdot \frac{dy}{dx} \right) + 2x = 0$ Here, we used the power rule and chain rule on $y^2$ to get $2y \cdot \frac{dy}{dx}$ .
We used the product rule on $3xy$ to get $3y + 3x \cdot \frac{dy}{dx}$ .
And the power rule on $x^2$ gives us $2x$ .
The derivative of the constant $7$ is just zero.

STEP 5

Now, let's isolate $\frac{dy}{dx}$ terms on one side and everything else on the other: $2y \cdot \frac{dy}{dx} - 3x \cdot \frac{dy}{dx} = 3y - 2x$

STEP 6

We can factor out $\frac{dy}{dx}$ : $\frac{dy}{dx}(2y - 3x) = 3y - 2x$

STEP 7

Finally, we divide both sides by $(2y - 3x)$ to solve for $\frac{dy}{dx}$ : $\frac{dy}{dx} = \frac{3y - 2x}{2y - 3x}$

STEP 8

So, our final answer is $\frac{dy}{dx} = \frac{3y - 2x}{2y - 3x}$ , which matches option (b)!

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Question7. Find $\frac{d y}{d x}$ if $y^{2}-3 x y+x^{2}=7$ . (a) $\frac{2 x+y}{3 x-2 y}$ (b) $\frac{3 y-2 x}{2 y-3 x}$ (c) $\frac{2 x}{3-2 y}$ (d) $\frac{2 x}{y}$ (e) None of these

Studdy Solution

STEP 1

STEP 2

1. Implicit Differentiation
2. Isolate and Simplify

STEP 3

Let's differentiate both sides of the equation $y^2 - 3xy + x^2 = 7$ with respect to $x$ .
Remember, we're thinking of $y$ as a function of $x$ , so we'll need the chain rule for terms involving $y$ .

STEP 4

STEP 5

Now, let's isolate $\frac{dy}{dx}$ terms on one side and everything else on the other: $2y \cdot \frac{dy}{dx} - 3x \cdot \frac{dy}{dx} = 3y - 2x$

STEP 6

We can factor out $\frac{dy}{dx}$ : $\frac{dy}{dx}(2y - 3x) = 3y - 2x$

STEP 7

Finally, we divide both sides by $(2y - 3x)$ to solve for $\frac{dy}{dx}$ : $\frac{dy}{dx} = \frac{3y - 2x}{2y - 3x}$

STEP 8

So, our final answer is $\frac{dy}{dx} = \frac{3y - 2x}{2y - 3x}$ , which matches option (b)!

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Question7. Find dydx\frac{d y}{d x}dxdy​ if y2−3xy+x2=7y^{2}-3 x y+x^{2}=7y2−3xy+x2=7. (a) 2x+y3x−2y\frac{2 x+y}{3 x-2 y}3x−2y2x+y​ (b) 3y−2x2y−3x\frac{3 y-2 x}{2 y-3 x}2y−3x3y−2x​ (c) 2x3−2y\frac{2 x}{3-2 y}3−2y2x​ (d) 2xy\frac{2 x}{y}y2x​ (e) None of these

Studdy Solution

STEP 1

STEP 2

1. Implicit Differentiation2. Isolate and Simplify

STEP 3

Let's **differentiate** both sides of the equation y2−3xy+x2=7y^2 - 3xy + x^2 = 7y2−3xy+x2=7 with respect to xxx.Remember, we're thinking of yyy as a function of xxx, so we'll need the **chain rule** for terms involving yyy.

STEP 4

STEP 5

Now, let's **isolate** dydx\frac{dy}{dx}dxdy​ terms on one side and everything else on the other: 2y⋅dydx−3x⋅dydx=3y−2x 2y \cdot \frac{dy}{dx} - 3x \cdot \frac{dy}{dx} = 3y - 2x 2y⋅dxdy​−3x⋅dxdy​=3y−2x

STEP 6

We can **factor out** dydx\frac{dy}{dx}dxdy​: dydx(2y−3x)=3y−2x \frac{dy}{dx}(2y - 3x) = 3y - 2x dxdy​(2y−3x)=3y−2x

STEP 7

Finally, we **divide** both sides by (2y−3x)(2y - 3x)(2y−3x) to solve for dydx\frac{dy}{dx}dxdy​: dydx=3y−2x2y−3x \frac{dy}{dx} = \frac{3y - 2x}{2y - 3x} dxdy​=2y−3x3y−2x​

STEP 8

So, our **final answer** is dydx=3y−2x2y−3x\frac{dy}{dx} = \frac{3y - 2x}{2y - 3x}dxdy​=2y−3x3y−2x​, which matches option (b)!

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Question7. Find $\frac{d y}{d x}$ if $y^{2}-3 x y+x^{2}=7$ . (a) $\frac{2 x+y}{3 x-2 y}$ (b) $\frac{3 y-2 x}{2 y-3 x}$ (c) $\frac{2 x}{3-2 y}$ (d) $\frac{2 x}{y}$ (e) None of these

1. Implicit Differentiation
2. Isolate and Simplify

Let's differentiate both sides of the equation $y^2 - 3xy + x^2 = 7$ with respect to $x$ .
Remember, we're thinking of $y$ as a function of $x$ , so we'll need the chain rule for terms involving $y$ .

Now, let's isolate $\frac{dy}{dx}$ terms on one side and everything else on the other: $2y \cdot \frac{dy}{dx} - 3x \cdot \frac{dy}{dx} = 3y - 2x$

We can factor out $\frac{dy}{dx}$ : $\frac{dy}{dx}(2y - 3x) = 3y - 2x$

Finally, we divide both sides by $(2y - 3x)$ to solve for $\frac{dy}{dx}$ : $\frac{dy}{dx} = \frac{3y - 2x}{2y - 3x}$

So, our final answer is $\frac{dy}{dx} = \frac{3y - 2x}{2y - 3x}$ , which matches option (b)!