find the slope of the tangent line to the curve below at the point (7,4) square root of 4x+y + square root of 2xy= 13.14016902304

To find the slope of the tangent line to the curve given by:

\[
\sqrt{4x + y} + \sqrt{2xy} = 13.14016902304
\]

at the point \((7, 4)\), we need to differentiate the equation implicitly with respect to \(x\).

### Step 1: Differentiate the Equation

Let \( F(x, y) = \sqrt{4x + y} + \sqrt{2xy} \). We differentiate each part with respect to \(x\).

\[
F(x, y) = \sqrt{4x + y} + \sqrt{2xy}
\]

1. **Differentiate \(\sqrt{4x + y}\):**

\[
\frac{d}{dx}\left(\sqrt{4x + y}\right) = \frac{1}{2\sqrt{4x + y}} \cdot (4 + y')
\]

2. **Differentiate \(\sqrt{2xy}\):**

\[
\frac{d}{dx}\left(\sqrt{2xy}\right) = \frac{1}{2\sqrt{2xy}} \cdot (2y + 2xy')
\]

The derivative of \(F(x, y)\) becomes:

\[
\frac{1}{2\sqrt{4x + y}}(4 + y') + \frac{1}{2\sqrt{2xy}}(2y + 2xy') = 0
\]

### Step 2: Substitute \((7, 4)\) into the Derivative

1. Evaluate \(\sqrt{4(7) + 4} = \sqrt{32} = 4\sqrt{2}\).
   
2. Evaluate \(\sqrt{2(7)(4)} = \sqrt{56} = 2\sqrt{14}\).

Substituting these values:

\[
\frac{1}{2(4\sqrt{2})}(4 + y') + \frac{1}{2(2\sqrt{14})}(8 + 14y') = 0
\]

\[
\frac{1}{8\sqrt{2}}(4 + y') + \frac{1}{4\sqrt{14}}(8 + 14y') = 0
\]

### Step 3: Solve for \(y'\)

1. Multiply each term by \(8\sqrt{2}\) to eliminate the denominator of the first fraction:

\[
(4 + y') + \frac{8\sqrt{2}}{4\sqrt{14}}(8 + 14y') = 0
\]

2. Simplify \(\frac{8\sqrt{2}}{4\sqrt{14}} = \frac{2\sqrt{2}}{\sqrt{14}} = \frac{\sqrt{28}}{14} = \frac{\sqrt{7}}{7}\).

\[
(4 + y') + \frac{\sqrt{7}}{7}(8 + 14y') = 0
\]

3. Multiply through by 7 to clear the fraction:

\[
7(4 + y') + \sqrt{7}(8 + 14y') = 0
\]

\[
28 + 7y' + 8\sqrt{7} + 14\sqrt{7}y' = 0
\]

\[
(7 + 14\sqrt{7})y' + (28 + 8\sqrt{7}) = 0
\]

### Step 4: Solve for \(y'\)

\[
y' = -\frac{28 + 8\sqrt{7}}{7 + 14\sqrt{7}}
\]

This fraction represents the slope of the tangent line at \((7, 4)\).

Would you like further details or any clarification on the steps?

### Questions to Consider:
1. How do you calculate the slope of the tangent for other implicit functions?
2. What is implicit differentiation, and how does it work?
3. Why is it important to substitute the specific point before solving for \(y'\)?
4. How would the equation change if the point were different, say \((5, 3)\)?
5. What are some methods to simplify complex radicals in differentiation problems?

### Tip:
Always check that the function is differentiable at the point of interest before applying differentiation techniques.

Find the slope of the tangent line to the curve at the point (7,4): √(4x + y) + √(2xy) = 13.14016902304.

Learn how to find the slope of the tangent line to the curve √(4x + y) + √(2xy) = 13.14016902304 at the point (7,4) using implicit differentiation. This step-by-step solution covers key calculus concepts like the chain rule and differentiating radical functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation