find the derivative of ( f(x) ).\n\n( f(x)=9^{x}-7^{x} )\n\n( 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)