We are tasked with finding $\frac{dy}{dx}$ for the function given in the image.

The function is:

$y = e^{\frac{8}{9}x}$

Step 1: Identify $y = f(u)$ and $u = g(x)$

To rewrite this function in the form $y = f(u)$ and $u = g(x)$, we can set:

$u = \frac{8}{9}x$

This means the function for $y$ becomes:

$y = e^u$

Thus, we have $y = f(u)$ where $f(u) = e^u$ and $u = g(x)$ where $g(x) = \frac{8}{9}x$.

Step 2: Differentiate using the chain rule

We need to compute $\frac{dy}{dx}$, and we'll use the chain rule:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

• $\frac{dy}{du} = \frac{d}{du}(e^u) = e^u$
• $\frac{du}{dx} = \frac{d}{dx}\left(\frac{8}{9}x\right) = \frac{8}{9}$

Step 3: Combine results

Now, substitute the expressions into the chain rule formula:

$\frac{dy}{dx} = e^u \cdot \frac{8}{9}$

Substitute $u = \frac{8}{9}x$ back into the equation:

$\frac{dy}{dx} = e^{\frac{8}{9}x} \cdot \frac{8}{9}$

Thus, the derivative of the function is:

$\frac{dy}{dx} = \frac{8}{9}e^{\frac{8}{9}x}$

Would you like further explanation or have any questions?

Related Questions:

1. How do you apply the chain rule in more complex functions?
2. What is the derivative of a natural exponential function in general?
3. How can the chain rule be extended to higher-order derivatives?
4. How do you apply implicit differentiation when functions are not explicitly written in terms of $x$?
5. What is the significance of exponential functions in growth and decay problems?

Tip: When applying the chain rule, always differentiate the outer function first and then multiply by the derivative of the inner function.