# The Slope of The Slope: Second Derivatives

The second derivative measures a function's rate of curvature change, obtained by differentiating the first derivative. It's pivotal in determining concavity, inflection points, and optimizing real-world scenarios like motion dynamics.

The second derivatives create their own graph, which is distinct from the original function’s graph. This graph, often called the “second derivative graph” or “concavity graph,” visually represents the rate at which the slope of the original function changes. It helps in understanding the curvature behavior of the original function, indicating where it’s concave up or down, and locating any inflection points where the concavity changes.

Since you are already equipped with the tools required to solve the problems involving the second derivative,

let’s dive into problems:

$$\text{Find the first and second derivatives of } f(x) = x \sin x.$$

$$u = x, \ u’ = 1$$

$$v = \sin x, \ v’ = \cos x$$

$$f'(x) = 1 \cdot \sin x + x \cdot \cos x$$

$$f'(x) = \sin x + x \cos x$$

$$(\sin x)’ = \cos x$$

$$(x \cos x)’ = 1 \cdot \cos x + x \cdot (-\sin x)$$

$$(x \cos x)’ = \cos x – x \sin x$$

$$f”(x) = \cos x + \cos x – x \sin x$$

$$f”(x) = 2\cos x – x \sin x$$

And that’s the second derivative of $$xsin x$$

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