Question

If y = \(10^{10^x}\), then \(\frac{dy}{dx}\) is :

• A. \(10^{10^x}\) (log 10)
• B. \(10^{10^x}\) (log 10)²
• C. \(10^{10^x}\) 10^x (log 10)²
• D. \(10^{10^x}\) 10^x (log 10)

Answer 1

The Correct Option is C

To find the derivative of \( y = 10^{10^x} \) with respect to \( x \), we start by taking the natural logarithm on both sides: \( \ln y = \ln(10^{10^x}) \). By properties of logarithms, this simplifies to \( \ln y = 10^x \cdot \ln 10 \). Now, differentiate both sides with respect to \( x \):

\(\frac{d}{dx}(\ln y) = \frac{d}{dx}(10^x \cdot \ln 10)\)

Using the chain rule on the left and product rule on the right, we obtain:

\(\frac{1}{y} \frac{dy}{dx} = \ln 10 \cdot \frac{d}{dx}(10^x)\)

Since \( \frac{d}{dx}(10^x) = 10^x \cdot \ln 10 \), it follows:

\(\frac{1}{y} \frac{dy}{dx} = \ln 10 \cdot 10^x \cdot \ln 10\)

Simplifying yields:

\(\frac{1}{y} \frac{dy}{dx} = 10^x \cdot (\ln 10)^2\)

Multiplying both sides by \( y = 10^{10^x} \) gives:

\(\frac{dy}{dx} = 10^{10^x} \cdot 10^x \cdot (\ln 10)^2\)

Thus, the derivative \(\frac{dy}{dx}\) is:

\(10^{10^x} \cdot 10^x \cdot (\ln 10)^2\)