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

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

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

Answer

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})