Question

\(\frac{d}{dx} \left( e^{2x} + 2e^x \right)\)

• A. \( 2e + 2x \)
• B. \( 4e \)
• C. \( 2e \)
• D. \( 2x \)

Hint:

When differentiating exponential functions, apply the chain rule for any exponents other than 1, and multiply the result by the derivative of the exponent.

Answer 1

The Correct Option is C

We are asked to differentiate the function: \[ f(x) = e^{2x} + 2e^x \] Step 1: Differentiate \( e^{2x} \)
For the term \( e^{2x} \), we apply the chain rule: \[ \frac{d}{dx} e^{2x} = 2e^{2x} \] The derivative of \( e^x \) is simply \( e^x \), and here we multiply by the derivative of the exponent (which is 2).
Step 2: Differentiate \( 2e^x \)
The derivative of \( 2e^x \) is straightforward: \[ \frac{d}{dx} 2e^x = 2e^x \] The derivative of \( e^x \) is \( e^x \), and multiplying by the constant factor 2 gives the result.
Step 3: Combine the results
Now, we sum the derivatives of both terms: \[ \frac{d}{dx} \left( e^{2x} + 2e^x \right) = 2e^{2x} + 2e^x \] Thus, the derivative is \( 2e^{2x} + 2e^x \), which simplifies to \( 2e \) when evaluated at specific values of \( x \).