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$$\text { Find } f^{\prime \prime}(x) \text { for. (a) } f(x)=\frac{\left(x^{2}+1\right)^{2}}{\left(x^{2}-1\right)^{3}} \text { (b) } f(x)=\left(x^{2}+1\right)^{10}$$

(a) $\frac{2\left(3 x^{2}+5\right)\left(x^{4}+10 x^{2}+1\right)}{\left(x^{2}-1\right)^{5}}$(b) $20\left(x^{2}+1\right)^{8}\left(19 x^{2}+1\right)$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Campbell University

Harvey Mudd College

Baylor University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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We want to find out double primer backs the second derivative of X. Or rather the second derivative of X. For A and B below for a We have F equals X squared plus one spirit over X squared minus one cube to find a suitable primary. First have to find a crime and then to differentiate again. So to find F prime, we use the quotient rule. So F prime is equal to G F prime minus F G prime over G squared. Using algebraic methods, we can simplify this to show negative two X times X squared plus one times X plus five over. Experiment is one of the fourth. That's we find F double prime using the exact same method of the quotient rule and this we obtained after the prime equals two times the experience five X to the fourth plus A squared plus one over X squared minus one of the fifth. Next to be we have experienced one of the 10th for this. We use the chain rule. F prime equals 10 times expert was one of the nine times two X or 20 x x plus one of the nine to find a suitable primary. Use the product rule as well as the general. This gives F double prime equals 20 times experience, one of the nine for 360 x squared x squared plus one of the eighth.

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