Given 3x²+2xy-5y²=0 find dy by dx at point (1,1)

To find \(\frac{dy}{dx}\) from the given equation \(3x^2 + 2xy - 5y^2 = 0\), we will use **implicit differentiation**.

### Step 1: Differentiate both sides of the equation with respect to \(x\).
- Differentiate \(3x^2\):  
  \(\frac{d}{dx}(3x^2) = 6x\)
  
- Differentiate \(2xy\) using the **product rule** (\(\frac{d}{dx}(xy) = x \frac{dy}{dx} + y\)):
  \[\frac{d}{dx}(2xy) = 2 \left( x \frac{dy}{dx} + y \right)\]

- Differentiate \(-5y^2\) (using the **chain rule**, as \(y\) is a function of \(x\)):
  \(\frac{d}{dx}(-5y^2) = -10y \frac{dy}{dx}\)

Now, applying these, we get the differentiated equation:

\[
6x + 2(x \frac{dy}{dx} + y) - 10y \frac{dy}{dx} = 0
\]

### Step 2: Substitute the point \((1,1)\) into the equation.
At \(x = 1\) and \(y = 1\), the equation becomes:

\[
6(1) + 2(1 \cdot \frac{dy}{dx} + 1) - 10(1) \frac{dy}{dx} = 0
\]
Simplifying:

\[
6 + 2(\frac{dy}{dx} + 1) - 10 \frac{dy}{dx} = 0
\]
\[
6 + 2 \frac{dy}{dx} + 2 - 10 \frac{dy}{dx} = 0
\]
\[
8 - 8 \frac{dy}{dx} = 0
\]

### Step 3: Solve for \(\frac{dy}{dx}\).
\[
8 = 8 \frac{dy}{dx}
\]
\[
\frac{dy}{dx} = 1
\]

Thus, \(\frac{dy}{dx} = 1\) at the point \((1,1)\).

Would you like further details or have any questions? 

Here are 5 related questions:
1. How is implicit differentiation different from normal differentiation?
2. What is the product rule, and why was it used here?
3. Can we find \(\frac{dy}{dx}\) at other points on the curve?
4. How would the chain rule be applied for other implicit equations?
5. What would the slope of the tangent line be at other points?

**Tip:** When using implicit differentiation, always apply the product and chain rules carefully when both variables are involved in a term.

Given 3x² + 2xy - 5y² = 0, find dy/dx at point (1,1).

Learn how to apply implicit differentiation to find dy/dx for the equation 3x² + 2xy - 5y² = 0 at the point (1,1). This solution uses key calculus concepts, including the product rule and chain rule, suitable for high school or early college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation