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Higher-order Derivatives

The first derivative is the slope of a curve, and the second derivative is the slope of the slope, like acceleration. So what's the third derivative? (Fun fact: it's actually called jerk.)

Second 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 \(f(x)\).

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