The equation in the image is as follows:

$\text{If } 2^x + 2^y = 2^{x + y}, \text{ then } \frac{dy}{dx} \text{ at } x = y = 1 \text{ is }$

To solve this problem, we can follow these steps:

Step 1: Differentiate the equation implicitly

The given equation is: $2^x + 2^y = 2^{x + y}$

Now differentiate both sides with respect to $x$. We'll use the chain rule, and recall that the derivative of $2^u$ with respect to $u$ is $2^u \ln(2)$.

$\frac{d}{dx}(2^x) + \frac{d}{dx}(2^y) = \frac{d}{dx}(2^{x + y})$

Breaking this down:

• The derivative of $2^x$ is $2^x \ln(2)$.
• The derivative of $2^y$ with respect to $x$ is $2^y \ln(2) \cdot \frac{dy}{dx}$ (chain rule, since $y$ depends on $x$).
• The derivative of $2^{x + y}$ is $2^{x + y} \ln(2) \cdot (1 + \frac{dy}{dx})$, applying the product rule and chain rule.

This gives us the equation:

$2^x \ln(2) + 2^y \ln(2) \frac{dy}{dx} = 2^{x + y} \ln(2) \left(1 + \frac{dy}{dx}\right)$

Step 2: Simplify the equation

Canceling out $\ln(2)$ from all terms:

$2^x + 2^y \frac{dy}{dx} = 2^{x + y} \left(1 + \frac{dy}{dx}\right)$

Step 3: Substitute $x = y = 1$

Now we substitute $x = y = 1$. This simplifies the exponents:

• $2^x = 2^1 = 2$
• $2^y = 2^1 = 2$
• $2^{x + y} = 2^{1 + 1} = 2^2 = 4$

The equation becomes:

$2 + 2 \frac{dy}{dx} = 4 \left(1 + \frac{dy}{dx}\right)$

Step 4: Solve for $\frac{dy}{dx}$

Expanding the right-hand side:

$2 + 2 \frac{dy}{dx} = 4 + 4 \frac{dy}{dx}$

Now, collect like terms:

$2 - 4 = 4 \frac{dy}{dx} - 2 \frac{dy}{dx}$

$-2 = 2 \frac{dy}{dx}$

Finally, solve for $\frac{dy}{dx}$:

$\frac{dy}{dx} = -1$

Answer:

$\boxed{-1}$

Do you need more details or have any questions? Here are five related questions:

1. What is implicit differentiation, and when is it used?
2. How do you differentiate exponential functions like $2^x$?
3. Why does the chain rule apply when differentiating with respect to $x$ for terms involving $y$?
4. Can this problem be solved using logarithmic differentiation?
5. How do the initial conditions $x = y = 1$ simplify the calculation?

Tip: When differentiating complex functions, breaking down the expression step by step and applying relevant rules such as the chain rule or product rule is essential for accuracy.