Question

find the derivative of ( f(x) ).

( f(x)=9^{x}-7^{x} )

( f^{prime}(x)= )

Explanation:

Step1: Differentiate \(9^x\)

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

Step2: Differentiate \(7^x\)

For \(y = 7^x\), using the formula \(\frac{d}{dx}(a^x)=a^x\ln a\), we have \(\frac{d}{dx}(7^x)=7^x\ln 7\)

Step3: Apply the difference rule

The difference rule \(\frac{d}{dx}(u - v)=\frac{du}{dx}-\frac{dv}{dx}\). Here \(u = 9^x\) and \(v = 7^x\). So \(f'(x)=\frac{d}{dx}(9^x)-\frac{d}{dx}(7^x)\)

Answer:

\(9^x\ln 9-7^x\ln 7\)