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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.)

# Higher Order Derivatives Problem Solving

Let $$f(t)=21 \ln (t+1)+t$$ be the position at time $$t$$ of point $$P$$ moving along a number line. What is the acceleration of $$P$$ when the speed is $$4?$$

Given function $$f(x)$$, let $$f^{(n)}(x)$$ be the $$n^{th}$$ derivative of the function $$f(x)$$ for any positive integer $$n$$. If $$f(x)=e^x+\cos x,$$ what is the value of $\sum_{k=1}^{124} f^{(k)}(0)?$

Let $$P$$ be a point moving in the $$xy$$-plane whose coordinates at time $$t$$ are given by $x(t)=2e^t\sin t, y(t)=6\cos t.$ What is the minimum value of the magnitude of the acceleration of $$P?$$

Consider the function $$f(x)=x^2e^{-x}$$. Define another function $$y=e^xf'(x)$$. If there exist constants $$a$$ and $$b$$ that satisfy $x^2y''-axy'+by=0$ for all real numbers $$x$$, what is the value of $$a+b$$?

$$f(x)$$ is a twice differentiable function such that $$f(2x) = f(x+5) + (x-5)^2$$. If $$f''(10) = \frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers, what is the value of $$a+b$$?

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