Question

Find the second derivative.
f(x)= 7e^(-x)-9e^(7x)

Answer 1

\[\large f(x)=7e^{-x}-9e^{7x}\]Hmm so we remember the derivative of \(e^x\) yes?
It gives us back the same thing.
\[\large (e^x)'=e^x\]The problem we have here though is that the exponent contains more than just \(x\). So we need to apply the chain rule when we take these derivatives, Multiplying by the derivative of the exponent.

Taking the first derivative,\[\large f'(x)=\color{royalblue}{7e^{-x}}(-x)'\color{royalblue}{-9e^{7x}}(7x)'\]We've taken the derivative of the blue parts already, they gave us the same thing back. The chain rule is telling us to multiply by the derivative of the exponent.
The little prime is to show that we still need to take it's derivative.\[\large f'(x)=\color{royalblue}{7e^{-x}}(-1)\color{royalblue}{-9e^{7x}}(7)\]Which simplifies to,\[\large f'(x)=-7e^{-x}-63e^{7x}\]

Understand how we were able to get the first derivative? Any confusion due to the constant coefficients in front of each term?