Differentiate w.r.t x
`e^(5x)`

Answer

To differentiate the function \( y = e^{5x} \) with respect to \( x \), we will use the chain rule. Here’s a step-by-step solution:

### Step 1: Identify the function
We start with the function:
\[ y = e^{5x} \]

### Step 2: Apply the chain rule
The chain rule states that if you have a composite function \( y = f(g(x)) \), then the derivative \( \frac{dy}{dx} \) is given by:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
In our case, let:
- \( f(u) = e^u \) where \( u = 5x \)
- Thus, \( g(x) = 5x \)

### Step 3: Differentiate the outer function
The derivative of \( f(u) = e^u \) with respect to \( u \) is:
\[ f'(u) = e^u \]
Now substituting back \( u = 5x \):
\[ f'(g(x)) = e^{5x} \]

### Step 4: Differentiate the inner function
Next, we differentiate the inner function \( g(x) = 5x \):
\[ g'(x) = 5 \]

### Step 5: Combine the results
Now, applying the chain rule:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) = e^{5x} \cdot 5 \]
Thus, we can write:
\[ \frac{dy}{dx} = 5e^{5x} \]

### Final Result
The derivative of \( y = e^{5x} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 5e^{5x} \]

---