If \(y = 10^{10^{x}}\), then \(\displaystyle\frac{dy}{dx}\) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(10^{10^{x}}\,10^{x}(\log 10)^2\)
The standard result is
\( \displaystyle \frac{d}{du}\left(a^{u}\right) = a^{u}\log a \)
Here the power \(u = 10^{x}\) is itself a function of x, so the chain rule is used twice:
\( \displaystyle \frac{dy}{dx} = 10^{10^{x}}\log 10 \times \frac{d}{dx}\left(10^{x}\right) \)
\( \displaystyle = 10^{10^{x}}\log 10 \times 10^{x}\log 10 \)
\( \displaystyle = 10^{10^{x}}\,10^{x}(\log 10)^2 \)
Why the other options are incorrect
Option (1) \(10^{10^{x}}(\log 10)\) — Incorrect. This treats the power \(10^{x}\) as a constant and leaves out its derivative \(10^{x}\log 10\) altogether.
Option (2) \(10^{10^{x}}(\log 10)^2\) — Incorrect. The factor \(10^{x}\) from the derivative of the power is missing.
Option (4) \(10^{10^{x}}\,10^{x}(\log 10)\) — Incorrect. Each of the two exponential steps gives a factor log 10, so log 10 must appear squared.