If ` y=e^(ax) ,then (dy)/(dx) =`

Options

A. ` -log y`
B. ` log y `
C. ` -ylog y `
D. ` ylog y `

Answer

To find the derivative of the function \( y = e^{ax} \) with respect to \( x \), we will follow these steps:

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

### Step 2: Differentiate with respect to \( x \)
To find \( \frac{dy}{dx} \), we will use the chain rule of differentiation. The chain rule states that if you have a composite function, the derivative is the derivative of the outer function multiplied by the derivative of the inner function.

Here, the outer function is \( e^u \) where \( u = ax \). The derivative of \( e^u \) with respect to \( u \) is \( e^u \), and the derivative of \( u = ax \) with respect to \( x \) is \( a \).

Thus, applying the chain rule:
\[ \frac{dy}{dx} = \frac{d}{dx}(e^{ax}) = e^{ax} \cdot \frac{d}{dx}(ax) \]
\[ \frac{dy}{dx} = e^{ax} \cdot a \]

### Step 3: Substitute back the expression for \( y \)
Since we know that \( y = e^{ax} \), we can substitute this back into our expression:
\[ \frac{dy}{dx} = a \cdot e^{ax} = a \cdot y \]

### Final Answer
Thus, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = ay \]

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