https://file.aiquickdraw.com/mathbot/file/1728677131927rqsdok3v.jpg

We are given the equation \(5x^2 + 2x + xy = 4\) and need to find \(y'(4)\) using implicit differentiation. Additionally, we know \(y(4) = -21\).

### Step 1: Implicit Differentiation of the Given Equation
Differentiate both sides of the equation \(5x^2 + 2x + xy = 4\) with respect to \(x\), keeping in mind that \(y\) is a function of \(x\) (so we apply the product rule for the \(xy\) term).

\[
\frac{d}{dx}(5x^2 + 2x + xy) = \frac{d}{dx}(4)
\]

Let's differentiate each term:

- \( \frac{d}{dx}(5x^2) = 10x \)
- \( \frac{d}{dx}(2x) = 2 \)
- For the \(xy\) term, we use the product rule: 
  \[
  \frac{d}{dx}(xy) = y \cdot \frac{d}{dx}(x) + x \cdot \frac{d}{dx}(y) = y + x \cdot y'
  \]

So, the differentiation of the left-hand side becomes:

\[
10x + 2 + y + x \cdot y' = 0
\]

### Step 2: Substitute \(x = 4\) and \(y(4) = -21\)
Now, substitute \(x = 4\) and \(y(4) = -21\) into the equation:

\[
10(4) + 2 + (-21) + 4 \cdot y' = 0
\]

This simplifies to:

\[
40 + 2 - 21 + 4 \cdot y' = 0
\]

\[
21 + 4y' = 0
\]

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

\[
4y' = -21
\]

\[
y' = -\frac{21}{4}
\]

Thus, \(y'(4) = -\frac{21}{4}\).

---

Would you like further explanation on any step, or have additional questions?

Here are 5 related questions for further practice:
1. How do you apply implicit differentiation to an equation involving trigonometric functions?
2. What is the product rule and how is it used in implicit differentiation?
3. How would the result change if \(y(4)\) were a different value?
4. Can implicit differentiation be used to solve for higher-order derivatives? If so, how?
5. How do we verify the result of implicit differentiation?

**Tip:** Always remember to apply the product rule when differentiating products of functions implicitly.

If 5x^2 + 2x + xy = 4 and y(4) = -21, find y'(4) by implicit differentiation.

Learn how to solve the implicit differentiation problem 5x^2 + 2x + xy = 4 with y(4) = -21. Step-by-step instructions using the product rule to find y'(4). Suitable for calculus 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