find the derivative of ( f(x) ).\n\n( f(x)=6^{x}+5^{x} )\n\n( f^{prime}(x)= )\n\nsubmit

Explanation:

Step1: Differentiate (6^x)

The derivative of (a^x) is (a^x\ln a). For (y = 6^x), using the formula (\frac{d}{dx}(a^x)=a^x\ln a), we have (\frac{d}{dx}(6^x)=6^x\ln 6).

Step2: Differentiate (5^x)

For (y = 5^x), using the formula (\frac{d}{dx}(a^x)=a^x\ln a), we get (\frac{d}{dx}(5^x)=5^x\ln 5).

Step3: Use the sum rule

If (f(x)=u(x)+v(x)), then (f^\prime(x)=u^\prime(x)+v^\prime(x)). Here (u(x) = 6^x) and (v(x)=5^x). So (f^\prime(x)=\frac{d}{dx}(6^x)+\frac{d}{dx}(5^x)).

Substituting the results from Step 1 and Step 2, we have (f^\prime(x)=6^x\ln 6 + 5^x\ln 5).

Answer:

(6^x\ln 6+5^x\ln 5)