Calculus
Higher-order Derivatives
Suppose that \[f(x)=xe^{ax+b}, f'(0)=3, f''(0)=6,\] where \(a\) and \(b\) are constants. What is \(a+b?\)
What is the second derivative of \(f(x)=x^{9}e^x\) with respect to \(x\)?
If \(\displaystyle f(x)=\frac{18}{x},\) what is the value of \[\lim_{x \to 1} \frac{f'(x)+18}{x-1}?\]
What is the second derivative of \[y=(x^2+2x+7)^2?\]
\(f(x)=xe^{ax+\ln b}\), where \(a\) and \(b\) are constants. If \(f'(0)=7\) and \(f''(0)=126\), what is the value of \(a+b\)?
Details and assumptions
\(\ln x\) denotes \(\log_e x\).
\(f''(x)\) denotes the second derivative of \(f(x)\).