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$$\text { Find } f^{\prime \prime}(x) \text { for } f(x)=(x-1) /(3 x+1)$$

$$\frac{-24}{(3 x+1)^{3}}$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Missouri State University

Baylor University

University of Nottingham

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.

01:03

Find $f^{\prime}(x)$$$…

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problems I would like you to find after the prime of X for F of X equals X minus one divided by three x plus one. Starting off Taking that first derivative, we can use the quotient rule here to say that F prime of X is equal to three x plus one times one minus x minus one times three all divided by three x minus one squared. You can then simplify that. Two three X plus one minus three X minus three All over three X minus one squared that is then equivalent to 4/3 X minus one squared, which is then equal to four times three X minus one to the negative, too. Now taking the second derivative F double prime of X of this expression here that would be negative. Eight if you bring down this too multiplied by the four three x minus one to the negative three multiplied by three. Using the chain role. Um and then from there that simplifies to negative 24 divided by three x minus one to the third and that is your answer to question five

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