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Use the definition of a derivative to find $ f'(x) $ and $ f''(x) $. Then graph $ f $, $ f' $, and $ f'' $ on a common screen and check to see if your answers are reasonable.

$ f(x) = 3x^2 + 2x + 1 $

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$6 x+2$ ; 6

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Oregon State University

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University of Nottingham

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

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.

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this problem number fifty three of this two. Re calculus. Eighth edition, Section two point eight. Use the definition of the derivative to find of prime of acceptable crime of X in graphic f f prime in if double prime in a common screen and check to see if your answers are reasonable. So, uh, let's start with that fax. Uh, this is to find as the limit as h approaches cereal of this function evaluated. Explore sage three times a quantity. Experts age squared plus two times the quantity X plus each plus one minus the function by itself three X squared plus two x plus one. Okay. And this is all the writer by h. Our next step is to expand the numerator little bit. This binomial squared expose Shin Quantity Square It is going to be a X squared plus two exchange pulsate squared multiplied by three will be three x squared plus six x age plus three h squared. Then we're going to multiply and distribute to here to expose to age. And then we have a plus one. And then in this part here, we're gonna distribute the negative negative three x squared negative two X uh, negative one. Um, this is all about about H. Now. We identified some terms that might be cancelled, such as three X squared and negative three X squared. Ah, positive to exit, *** Twix and Positive one. And make it for me. The remaining three terms six X age plus three squared plus two each. Have one h er in common so that can counsel with the agent the denominator. Therefore, we're left over with the limit as each purchase zero uh, six x plus three age plus two and his H approaches zero three h purchases. Hero answer. Here is three x Plus two six exposed to and that is our answer for a crime of X. Let's do the same for the next derivative story to an ex f double prime of X, and we're using this function now. The results interrogative as thie function to evaluate his limit with the limited age approaches zero where we do the same steps again just with a new function. The function is six x squared plus two and we're going to play an extra sage here and there were subtracting at six x plus two. Put this in some apprentices on the road by h came, an extent will be, too. Distribute the six in the numerator six thanks plus six each, plus two. And then there should be two negative negative. Six six minus two Battered by each. Think six, six and negative six will cancel, plus two in a little cancel. Then we're left with six h or age, which, if we simplified, becomes just six. That's the only remaining term. The limit as a chipper to zero of just the constant number six is equal to just six. So the second derivative we are is just six. Now we have, with the function, affects eyes to the X Corp was too explicit. One. The derivative of Prime of X, which is equaled at six. Extras, too. And the secondary of crime of double paramedics, which is equal to six. And we're going to take all three of these functions were evident. A common screen. A show in here I'm here. We have the function f of X to X weapons to express one in purple, the interpretive six seconds to shut and read linear function and then and the function for the derivative Secretariat of F Prime of X is equal to six. It's a constant. Now you have six, and this the's all make sense because starting with the original function F, it is decreasing here up until its minimum about here. Which makes sense because the linear function here's in the negative area. It's, ah, negative values all the way up until this point, hair where across that the X axis AP puts the function of his increasing. Therefore, the function six X Corp was too. Must be all positive afterward. And that's true. And then now we look at the relationship between Have prime and that's double prime, or F prime eyes. A linear function. Six x plus two. It's the slip of all that of its tensions line pearl. Equal attention lines here all constant would taunt slope, and that's left of six and then Isaac Exactly shown here about the derivative of a prime

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