Find the derivative of the given function.
$f(x)=2^x-5^x$

• A. $x^2\(\ln\)2-x^5\(\ln\)5$
• B. $2^{x}\(\ln\)2-5^{x}\(\ln\)5$
• C. $2^{x}\(\ln\) x-5^{x}\(\lnx\)$
• D. $2\(\lnx\)-5\(\ln\) x$

Answer: (B)

Verified step by step guidance

1. Identify the function for which you need to find the derivative: \( f(x) = 2^x - 5^x \).
2. Recall the derivative rule for exponential functions: \( \frac{d}{dx} a^x = a^x \ln a \), where \( a \) is a constant.
3. Apply the derivative rule to each term separately. For \( 2^x \), the derivative is \( 2^x \ln 2 \). For \( 5^x \), the derivative is \( 5^x \ln 5 \).
4. Combine the derivatives of each term to find the derivative of the entire function: \( f'(x) = 2^x \ln 2 - 5^x \ln 5 \).
5. Verify the result by checking each step and ensuring the application of the derivative rule is correct for exponential functions.