Question

find the derivative of ( f(x) ).
( f(x)=e^{-5 x}+9^{x} )
( f^{prime}(x)= )

Explanation:

Step1: Differentiate \(e^{-5x}\)

Use the chain rule \((e^{u})^\prime=e^{u}\cdot u^\prime\). Let \(u = - 5x\), then \(u^\prime=-5\). So \((e^{-5x})^\prime=e^{-5x}\cdot(-5)=-5e^{-5x}\)

Step2: Differentiate \(9^{x}\)

Use the formula \((a^{x})^\prime=a^{x}\ln a\). Here \(a = 9\), so \((9^{x})^\prime=9^{x}\ln9\)

Step3: Combine the derivatives

Since \(f(x)=e^{-5x}+9^{x}\), then \(f^\prime(x)=(e^{-5x})^\prime+(9^{x})^\prime\)

Answer:

\(-5e^{-5x}+9^{x}\ln9\)