Question

find the derivative of ( f(x) ).

( f(x)=6^{x}+5^{x} )

( f^{prime}(x)= )

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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:

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