3.1 Defining the derivative  (Page 2/10)

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In [link] (a) we see that, as the values of $x$ approach $a,$ the slopes of the secant lines provide better estimates of the rate of change of the function at $a.$ Furthermore, the secant lines themselves approach the tangent line to the function at $a,$ which represents the limit of the secant lines. Similarly, [link] (b) shows that as the values of $h$ get closer to $0,$ the secant lines also approach the tangent line. The slope of the tangent line at $a$ is the rate of change of the function at $a,$ as shown in [link] (c).

You can use this site to explore graphs to see if they have a tangent line at a point.

In [link] we show the graph of $f\left(x\right)=\sqrt{x}$ and its tangent line at $\left(1,1\right)$ in a series of tighter intervals about $x=1.$ As the intervals become narrower, the graph of the function and its tangent line appear to coincide, making the values on the tangent line a good approximation to the values of the function for choices of $x$ close to $1.$ In fact, the graph of $f\left(x\right)$ itself appears to be locally linear in the immediate vicinity of $x=1.$

Formally we may define the tangent line to the graph of a function as follows.

Definition

Let $f\left(x\right)$ be a function defined in an open interval containing $a.$ The tangent line to $f\left(x\right)$ at $a$ is the line passing through the point $\left(a,f\left(a\right)\right)$ having slope

${m}_{\text{tan}}=\underset{x\to a}{\text{lim}}\frac{f\left(x\right)-f\left(a\right)}{x-a}$

provided this limit exists.

Equivalently, we may define the tangent line to $f\left(x\right)$ at $a$ to be the line passing through the point $\left(a,f\left(a\right)\right)$ having slope

${m}_{\text{tan}}=\underset{h\to 0}{\text{lim}}\frac{f\left(a+h\right)-f\left(a\right)}{h}$

provided this limit exists.

Just as we have used two different expressions to define the slope of a secant line, we use two different forms to define the slope of the tangent line. In this text we use both forms of the definition. As before, the choice of definition will depend on the setting. Now that we have formally defined a tangent line to a function at a point, we can use this definition to find equations of tangent lines.

Finding a tangent line

Find the equation of the line tangent to the graph of $f\left(x\right)={x}^{2}$ at $x=3.$

First find the slope of the tangent line. In this example, use [link] .

$\begin{array}{ccccc}\hfill {m}_{\text{tan}}& =\underset{x\to 3}{\text{lim}}\frac{f\left(x\right)-f\left(3\right)}{x-3}\hfill & & & \text{Apply the definition.}\hfill \\ & =\underset{x\to 3}{\text{lim}}\frac{{x}^{2}-9}{x-3}\hfill & & & \text{Substitute}\phantom{\rule{0.2em}{0ex}}f\left(x\right)={x}^{2}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}f\left(3\right)=9.\hfill \\ & =\underset{x\to 3}{\text{lim}}\frac{\left(x-3\right)\left(x+3\right)}{x-3}=\underset{x\to 3}{\text{lim}}\left(x+3\right)=6\hfill & & & \text{Factor the numerator to evaluate the limit.}\hfill \end{array}$

Next, find a point on the tangent line. Since the line is tangent to the graph of $f\left(x\right)$ at $x=3,$ it passes through the point $\left(3,f\left(3\right)\right).$ We have $f\left(3\right)=9,$ so the tangent line passes through the point $\left(3,9\right).$

Using the point-slope equation of the line with the slope $m=6$ and the point $\left(3,9\right),$ we obtain the line $y-9=6\left(x-3\right).$ Simplifying, we have $y=6x-9.$ The graph of $f\left(x\right)={x}^{2}$ and its tangent line at $3$ are shown in [link] .

The slope of a tangent line revisited

Use [link] to find the slope of the line tangent to the graph of $f\left(x\right)={x}^{2}$ at $x=3.$

The steps are very similar to [link] . See [link] for the definition.

$\begin{array}{ccccc}\hfill {m}_{\text{tan}}& =\underset{h\to 0}{\text{lim}}\frac{f\left(3+h\right)-f\left(3\right)}{h}\hfill & & & \text{Apply the definition.}\hfill \\ & =\underset{h\to 0}{\text{lim}}\frac{{\left(3+h\right)}^{2}-9}{h}\hfill & & & \text{Substitute}\phantom{\rule{0.2em}{0ex}}f\left(3+h\right)={\left(3+h\right)}^{2}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}f\left(3\right)=9.\hfill \\ & =\underset{h\to 0}{\text{lim}}\frac{9+6h+{h}^{2}-9}{h}\hfill & & & \text{Expand and simplify to evaluate the limit.}\hfill \\ & =\underset{h\to 0}{\text{lim}}\frac{h\left(6+h\right)}{h}=\underset{h\to 0}{\text{lim}}\left(6+h\right)=6\hfill \end{array}$

We obtained the same value for the slope of the tangent line by using the other definition, demonstrating that the formulas can be interchanged.

why n does not equal -1
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
proof the formula integration of udv=uv-integration of vdu.?
Find derivative (2x^3+6xy-4y^2)^2
no x=2 is not a function, as there is nothing that's changing.
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
y=800
800
Bg
how do u factor the numerator?
Nonsense, you factor numbers
Antonio
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
volume between cone z=√(x^2+y^2) and plane z=2
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
How do we find the horizontal asymptote of a function using limits?
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
what is a function? f(x)
one that is one to one, one that passes the vertical line test
Andrew
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
Antonio
is x=2 a function?
The
restate the problem. and I will look. ty
is x=2 a function?
The
What is limit
it's the value a function will take while approaching a particular value
Dan
don ger it
Jeremy
what is a limit?
Dlamini
it is the value the function approaches as the input approaches that value.
Andrew
Thanx
Dlamini
Its' complex a limit It's a metrical and topological natural question... approaching means nothing in math
Antonio
is x=2 a function?
The
3y^2*y' + 2xy^3 + 3y^2y'x^2 = 0 sub in x = 2, and y = 1, isolate y'
what is implicit of y³+x²y³=5 at (2,1)
tel mi about a function. what is it?
Jeremy
A function it's a law, that for each value in the domaon associate a single one in the codomain
Antonio
function is a something which another thing depends upon to take place. Example A son depends on his father. meaning here is the father is function of the son. let the father be y and the son be x. the we say F(X)=Y.
Bg
yes the son on his father
pascal
a function is equivalent to a machine. this machine makes x to create y. thus, y is dependent upon x to be produced. note x is an independent variable
moe
x or y those not matter is just to represent.
Bg