Question

let ( y = 3^{x} ).
find ( \frac{d^{2}y}{dx^{2}} ).
( \frac{d^{2}y}{dx^{2}}=)

Explanation:

Step1: Find the first - derivative

Use the formula for the derivative of \(a^{x}\), which is \(\frac{d}{dx}(a^{x})=a^{x}\ln a\).
For \(y = 3^{x}\), the first - derivative \(\frac{dy}{dx}=3^{x}\ln 3\).

Step2: Find the second - derivative

Differentiate \(\frac{dy}{dx}=3^{x}\ln 3\) with respect to \(x\) again.
Since \(\ln 3\) is a constant, and using the formula \(\frac{d}{dx}(a^{x})=a^{x}\ln a\) (here \(a = 3\)), we have \(\frac{d^{2}y}{dx^{2}}=\ln 3\times\frac{d}{dx}(3^{x})\).
Substitute \(\frac{d}{dx}(3^{x}) = 3^{x}\ln 3\) into the above formula, then \(\frac{d^{2}y}{dx^{2}}=3^{x}(\ln 3)^{2}\).

Answer:

\(3^{x}(\ln 3)^{2}\)