derivative of `e^(x/a)`

Answer

To find the derivative of the function \( f(x) = e^{\frac{x}{a}} \), we can follow these steps:

### Step 1: Identify the function
We have the function:
\[ f(x) = e^{\frac{x}{a}} \]

### Step 2: Apply the chain rule
To differentiate \( f(x) \), we will use the chain rule. The chain rule states that if you have a composite function \( g(h(x)) \), then the derivative is given by:
\[ g'(h(x)) \cdot h'(x) \]
In our case, \( g(u) = e^u \) where \( u = \frac{x}{a} \).

### Step 3: Differentiate the outer function
The derivative of \( g(u) = e^u \) with respect to \( u \) is:
\[ g'(u) = e^u \]
Thus, substituting back for \( u \):
\[ g'\left(\frac{x}{a}\right) = e^{\frac{x}{a}} \]

### Step 4: Differentiate the inner function
Next, we differentiate the inner function \( h(x) = \frac{x}{a} \):
\[ h'(x) = \frac{1}{a} \]

### Step 5: Apply the chain rule
Now we can combine these results using the chain rule:
\[ f'(x) = g'\left(h(x)\right) \cdot h'(x) = e^{\frac{x}{a}} \cdot \frac{1}{a} \]

### Step 6: Write the final result
Therefore, the derivative of the function \( f(x) = e^{\frac{x}{a}} \) is:
\[ f'(x) = \frac{1}{a} e^{\frac{x}{a}} \]

### Summary of the solution:
The derivative of \( e^{\frac{x}{a}} \) is:
\[ f'(x) = \frac{1}{a} e^{\frac{x}{a}} \]

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