Differentiate this.
x^2 - sqrt(3)xy + 2y^2 = 5
Can someone show me?
use implicit differentiation.
2x -√3xdy/dx - √3y + 4ydy/dx = 0
dy/dx(4y - √3x) = √3y - 2x
dy/dx = (√3y - 2x)/(4y - √3x)
Certainly! To differentiate the given expression, we need to differentiate each term with respect to the variables involved.
Let's differentiate term by term:
1. Differentiating x^2:
To differentiate x^2, we use the power rule of differentiation. The power rule states that if we have a term of the form x^n, then its derivative with respect to x is nx^(n-1). Applying this rule to x^2, we get: d/dx (x^2) = 2x.
2. Differentiating -sqrt(3)xy:
Here, we have a product of three terms: -sqrt(3), x, and y.
- Differentiating -sqrt(3) = 0 (since it's a constant).
- Differentiating x = 1 (since the derivative of x with respect to x is 1).
- Differentiating y = 0 (since y doesn't involve x).
Therefore, the derivative of -sqrt(3)xy is 0.
3. Differentiating 2y^2:
Similar to the first term, we use the power rule for differentiation. The power rule applied to 2y^2 gives: d/dx (2y^2) = 4y.
So, to differentiate the entire expression x^2 - sqrt(3)xy + 2y^2, we add up the derivatives of each term:
d/dx (x^2 - sqrt(3)xy + 2y^2) = 2x + 0 + 4y
Therefore, the derivative of the given expression is 2x + 4y.