If \(y = 5^{2x}\), then \(\displaystyle\frac{dy}{dx}\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(2 \cdot 5^{2x}\log_e 5\)
For a constant base a, \(\frac{d}{du}a^{u} = a^{u}\log_e a\). Here \(u = 2x\), so the chain rule adds the factor \(\frac{du}{dx} = 2\):
\( \displaystyle \frac{dy}{dx} = 5^{2x}\log_e 5 \cdot 2 \)
\( = 2 \cdot 5^{2x}\log_e 5 \)
The power rule does not apply here: the variable is in the exponent, not in the base.