Given (2x+4y)^5 =x, find dy/dx in terms of x and y.

To find \(\frac{dy}{dx}\) for the given equation \((2x + 4y)^5 = x\), we will use implicit differentiation.

### Step 1: Differentiate both sides with respect to \(x\).
The given equation is:

\[
(2x + 4y)^5 = x
\]

Differentiating both sides with respect to \(x\):

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

### Step 2: Apply the chain rule to the left side.
Using the chain rule:

\[
5(2x + 4y)^4 \cdot \frac{d}{dx}(2x + 4y) = 1
\]

Now, differentiate \(2x + 4y\) with respect to \(x\):

\[
\frac{d}{dx}(2x + 4y) = 2 + 4\frac{dy}{dx}
\]

Substitute this back into the equation:

\[
5(2x + 4y)^4 \cdot (2 + 4\frac{dy}{dx}) = 1
\]

### Step 3: Solve for \(\frac{dy}{dx}\).
Expand and solve for \(\frac{dy}{dx}\):

\[
5(2x + 4y)^4 \cdot (2 + 4\frac{dy}{dx}) = 1
\]

Divide both sides by \(5(2x + 4y)^4\):

\[
2 + 4\frac{dy}{dx} = \frac{1}{5(2x + 4y)^4}
\]

Now, isolate \(\frac{dy}{dx}\):

\[
4\frac{dy}{dx} = \frac{1}{5(2x + 4y)^4} - 2
\]

Finally, solve for \(\frac{dy}{dx}\):

\[
\frac{dy}{dx} = \frac{1}{4} \left(\frac{1}{5(2x + 4y)^4} - 2\right)
\]

This is the derivative \(\frac{dy}{dx}\) in terms of \(x\) and \(y\).

---

Would you like more details on any of the steps, or do you have any questions? Here are some related questions to explore further:

1. How do you apply the chain rule in implicit differentiation?
2. What is implicit differentiation and when is it used?
3. Can you explain the importance of isolating \(\frac{dy}{dx}\) in differentiation problems?
4. What would change if the original equation were \((2x + 4y)^5 = x^2\)?
5. How does implicit differentiation differ from explicit differentiation?

**Tip:** Always keep track of your differentiation steps, especially when using the chain rule, to avoid mistakes in complex expressions.

Learn how to find the derivative dy/dx for the equation (2x + 4y)^5 = x using implicit differentiation. This problem covers implicit differentiation and the chain rule, suitable for advanced high school students or those studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation