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$f(x)=\{\begin{array}{c}9 x+5 \text { if } x \geq 1 \\ x^{2}+7 x+6 \text { if } x<1\end{array} \text { (a) What is } f^{\prime}(x) \text { if } x>1 ?$ (b) What is \right. $f^{\prime}(x)$ if $x<1 ?(\mathrm{c})$ What is the slope of the curve just to the left of $x=1 ?$ (d) What is $f^{\prime}(1) ?$

(a) Graph is answer(b) $2 x+7$(c) 9(d) 9

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 1

Slope of a Curve

Derivatives

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

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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here we are going to be differentiating parts of a piecewise function to start, we're gonna be looking at what f prime of X is if X is greater than one here taking our knowledge of piecewise functions, we know that that must be referring to the first part of the piecewise function where F of X is equal to nine X plus five. Because that's the piece for one X is greater than one taking the derivative of that we get that F prime of X is equal to nine doing the same thing. But for one X is less than one. We can see that we're gonna be using that second piece of our piecewise function where F of X is equal to X squared plus seven X plus six taking the derivative here, we find that F prime of X is equal to two X plus seven. Now we're taking a look at what must the slope of the curve just to the left of X equals one b. So at this point, if it is at X must be to the left of X equals one, we know that it must be less than one so we're gonna be using Our F of X is equal to X squared plus seven X plus six. And we just found the derivative of that to be equal to two X plus seven. Knowing that the derivative allows us to calculate the slope we can plug in for F prime of X, we can plug one in there, giving us f prime of one being equal to two times one plus seven. Here we end up finding that the derivative just to the left of X equals one is probably going to be about nine. And now down here for F prime of one is basically what we just did. We can go ahead and plug that value in, but instead of doing it for that bottom equation where it equals one or where it was less than one Now we're going to plug it back into our first equation where F of X is equal to nine x plus five, because that is when X is greater than or equal to one. So this is when we want to be using this one. We already found the derivative of this in part a to be equal to nine which means it must be that when f prime of one or when X is equal to one F prime would also be equal to nine.

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