Question

find the derivative of ( f(x) ).

( f(x)=9^{x}+e^{x} )

( f^{prime}(x)= )

Explanation:

Step1: Derivative of \(a^x\)

The derivative of \(a^x\) is \(a^x\ln a\). For \(y = 9^x\), using the formula \(\frac{d}{dx}(a^x)=a^x\ln a\) (where \(a = 9\)), we have \(\frac{d}{dx}(9^x)=9^x\ln 9\).

Step2: Derivative of \(e^x\)

The derivative of \(e^x\) is \(e^x\), i.e., \(\frac{d}{dx}(e^x)=e^x\).

Step3: Sum - rule of derivatives

If \(f(x)=u(x)+v(x)\), then \(f^{\prime}(x)=u^{\prime}(x)+v^{\prime}(x)\). Here \(u(x)=9^x\) and \(v(x)=e^x\). So \(f^{\prime}(x)=\frac{d}{dx}(9^x)+\frac{d}{dx}(e^x)\).

Answer:

\(9^x\ln 9 + e^x\)