Question

If \( y = e^{2x} \), then find \( \dfrac{dy}{dx} \).

• A. \( 2e^{2x} \)
• B. \( e^x \)
• C. \( 2xe^{2x} \)
• D. \( e^{2x}+2 \)

Hint:

For any exponential function of the form \( y = e^{f(x)} \), its derivative is always the original function multiplied by the derivative of the exponent.
That is, \( y' = f'(x) \cdot e^{f(x)} \).
This shortcut allows you to write down the answer instantly for any composite exponential function.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
In this problem, we are given a composite exponential function \( y = e^{2x} \) and asked to find its first derivative with respect to \( x \).
This requires applying the rules of differentiation for exponential functions.

Step 2: Key Formula or Approach:
We will use the Chain Rule of differentiation.
If \( y = f(u) \) and \( u = g(x) \), then:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
For an exponential function where \( y = e^{kx} \) (with \( k \) being a constant), the derivative formula is:
\[ \frac{d}{dx}(e^{kx}) = k \cdot e^{kx} \]

Step 3: Detailed Explanation:
Let us define our inner function as \( u = 2x \) and the outer function as \( y = e^u \).
First, we find the derivative of the inner function \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{d}{dx}(2x) = 2 \]
Second, we find the derivative of the outer function \( y \) with respect to \( u \):
\[ \frac{dy}{du} = \frac{d}{du}(e^u) = e^u \]
Now, we apply the Chain Rule to find \( \frac{dy}{dx} \) by multiplying these two derivatives:
\[ \frac{dy}{dx} = e^u \cdot 2 \]
Substituting back \( u = 2x \) into the equation, we get:
\[ \frac{dy}{dx} = 2e^{2x} \]
Let us analyze why the other options are incorrect:
- Option (B) \( e^x \) is incorrect because it completely ignores both the coefficient \( 2 \) in the exponent and the chain rule.
- Option (C) \( 2xe^{2x} \) is incorrect because it mistakenly applies the power rule to the exponent instead of keeping the exponential term intact.
- Option (D) \( e^{2x}+2 \) is incorrect because it adds the constant instead of multiplying by it.

Step 4: Final Answer:
Hence, the derivative of the given function is \( 2e^{2x} \), which corresponds to Option (A).