What is the dy/dx if x=-1 and y=-2, of the function 3x^2y - 4xy^2 = 10?
Understand the Problem
The question is asking to find the derivative dy/dx of the implicit function 3x^2y - 4xy^2 = 10 at the point where x=-1 and y=-2. This will involve using implicit differentiation to derive dy/dx and substituting the specified values to determine the slope of the function at that point.
Answer
The derivative \( \frac{dy}{dx} \) at the point \( (-1, -2) \) is \( \frac{4}{19} \).
Answer for screen readers
The derivative ( \frac{dy}{dx} ) at the point ( (-1, -2) ) is ( \frac{4}{19} ).
Steps to Solve
- Differentiate both sides implicitly
We start by differentiating both sides of the equation ( 3x^2y - 4xy^2 = 10 ) with respect to ( x ). Remember to use the product rule where necessary and the chain rule for the ( y ) terms.
Differentiating gives:
$$ \frac{d}{dx}(3x^2y) - \frac{d}{dx}(4xy^2) = 0 $$
Applying the product rule to both terms:
$$ 3(2xy + x^2\frac{dy}{dx}) - 4(y^2 + 2xy\frac{dy}{dx}) = 0 $$
- Combine and simplify the equation
Next, we simplify the derivative equation. Group the terms with ( \frac{dy}{dx} ) on one side and constant terms on the other side.
This leads to:
$$ 6xy + 3x^2\frac{dy}{dx} - 4y^2 - 8xy\frac{dy}{dx} = 0 $$
Then, rearranging the equation:
$$ (3x^2 - 8xy)\frac{dy}{dx} = 4y^2 - 6xy $$
- Solve for ( \frac{dy}{dx} )
We can now isolate ( \frac{dy}{dx} ):
$$ \frac{dy}{dx} = \frac{4y^2 - 6xy}{3x^2 - 8xy} $$
- Substitute the values ( x = -1 ) and ( y = -2 )
Now we will substitute ( x = -1 ) and ( y = -2 ) into the equation we found for ( \frac{dy}{dx} ):
Calculating this gives:
$$ \frac{dy}{dx} = \frac{4(-2)^2 - 6(-1)(-2)}{3(-1)^2 - 8(-1)(-2)} $$
- Calculate the value
Calculating the numerator and denominator separately:
Numerator:
$$ 4(4) - 12 = 16 - 12 = 4 $$
Denominator:
$$ 3(1) + 16 = 3 + 16 = 19 $$
So,
$$ \frac{dy}{dx} = \frac{4}{19} $$
The derivative ( \frac{dy}{dx} ) at the point ( (-1, -2) ) is ( \frac{4}{19} ).
More Information
This answer indicates the slope of the tangent line to the implicit function ( 3x^2y - 4xy^2 = 10 ) at the specific point ( (-1, -2) ). Implicit differentiation is often used when it is difficult to solve for ( y ) explicitly.
Tips
- Forgetting to apply the product rule correctly when differentiating terms involving both ( x ) and ( y ).
- Not simplifying the equation completely before solving for ( \frac{dy}{dx} .
- Miscalculating when substituting values, especially with signs.
