Question

find the derivative of ( f(x) ).

( f(x)=e^{-9 x}+9^{x} )

( f^{prime}(x)= )

Explanation:

Step1: Differentiate \(e^{-9x}\)

Use the chain rule. Let \(u = -9x\), then \(\frac{d}{dx}(e^u)=e^u\cdot\frac{du}{dx}\). Here \(\frac{du}{dx}=-9\), so \(\frac{d}{dx}(e^{-9x})=-9e^{-9x}\).

Step2: Differentiate \(9^x\)

Use the formula \(\frac{d}{dx}(a^x)=a^x\ln a\). For \(a = 9\), \(\frac{d}{dx}(9^x)=9^x\ln 9\).

Answer:

\(-9e^{-9x}+9^x\ln 9\)