`f(x) = cos(x^2), (0,1)` Evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result.

Textbook Question

Chapter 2, 2.4 - Problem 93 - Calculus of a Single Variable (10th Edition, Ron Larson).
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tiburtius | High School Teacher | (Level 2) Educator

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We will need to use chain ruleĀ 

`(f(g(x)))'=f'(g(x))cdot g'(x)`

First we find the first derivative.

`f'(x)=-sin(x^2)cdot 2x=-2x sin(x^2)`

Now we calculate second derivative, but for that we will need to use the following formula

`(f cdot g)'=f' cdot g+f cdot g'`

`f''(x)=-2sin(x^2)-2xcos(x^2)cdot 2x`

`f''(x)=-2sin(x^2)-4x^2cos(x^2)`

Now we evaluate the second derivative at 0.

`f''(0)=-2sin(0^2)-4cdot0^2cos(0^2)=0`

In the image below blue is the function and red is its second derivative.

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