# 3.1 Defining the derivative  (Page 2/10)

 Page 2 / 10 We can calculate the slope of a secant line in either of two ways.

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). The secant lines approach the tangent line (shown in green) as the second point approaches the first.

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 $x=1.$ For values of x close to 1 , the graph of f ( x ) = x and its tangent line appear to coincide.

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 tangent line to f ( x ) at x = 3 .

## 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.

#### Questions & Answers

find the domain and range of f(x)= 4x-7/x²-6x+8
Nick Reply
find the range of f(x)=(x+1)(x+4)
Jane Reply
-1, -4
Marcia
That's domain. The range is [-9/4,+infinity)
Jacob
If you're using calculus to find the range, you have to find the extrema through the first derivative test and then substitute the x-value for the extrema back into the original equation.
Jacob
Good morning,,, how are you
Harrieta Reply
d/dx{1/y - lny + X^3.Y^5}
mogomotsi Reply
How to identify domain and range
Umar Reply
hello
Akpevwe
He,,
Harrieta
hi
Dr
hello
velocity
I only talk to girls
Dr
women are smart then guys
Dr
Smarter
Adri
sorry
Dr
hi adri ana
Dr
:(
Shun
was up
Dr
hello
Adarsh
is it chatting app?.. I do not see any calculus here. lol
Adarsh
Find the arc length of the graph of f(x) = In (sinx) on the interval [Π/4, Π/2].
mukul Reply
Sand falling freely from a lorry form a conical shape whose height is always equal to one-third the radius of the base. a. How fast is the volume increasing when the radius of the base is (1m) and increasing at the rate of 1/4cm/sec Pls help me solve
ade
show that lim f(x) + lim g(x)=m+l
BARNABAS Reply
list the basic elementary differentials
Chio Reply
Differentiation and integration
Okikiola Reply
yes
Damien
proper definition of derivative
Syed Reply
the maximum rate of change of one variable with respect to another variable
Amdad
terms of an AP is 1/v and the vth term is 1/u show that the sum of uv terms is 1/2(uv+1)
Inembo Reply
what is calculus?
BISWAJIT Reply
calculus is math that studies the change in math, such as the rate and distance,
Tamarcus
what are the topics in calculus
Augustine
what is limit of a function?
Geoffrey Reply
what is x and how x=9.1 take?
Pravin Reply
what is f(x)
Inembo Reply
the function at x
Marc
also known as the y value so I could say y=2x or f(x)= 2x same thing just using functional notation your next question is what is dependent and independent variables. I am Dyslexic but know math and which is which confuses me. but one can vary the x value while y depends on which x you use. also
Marc
up domain and range
Marc
enjoy your work and good luck
Marc
I actually wanted to ask another questions on sets if u dont mind please?
Inembo
I have so many questions on set and I really love dis app I never believed u would reply
Inembo
Hmm go ahead and ask you got me curious too much conversation here
Adri
am sorry for disturbing I really want to know math that's why *I want to know the meaning of those symbols in sets* e.g n,U,A', etc pls I want to know it and how to solve its problems
Inembo
and how can i solve a question like dis *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
next questions what do dy mean by (A' n B^c)^c'
Inembo
The sets help you to define the function. The function is like a magic box where you put inside stuff(numbers or sets) and you get out the stuff but in different shapes (forms).
Adri
I dont understand what you wanna say by (A' n B^c)^c'
Adri
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Adri
Ok so the set is formed by vectors and not numbers
Adri
A vector of length n
Adri
But you can make a set out of matrixes as well
Adri
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
Wait what's your math level?
Adri
High-school?
Adri
yes
Inembo
am having big problem understanding sets more than other math topics
Inembo
So f:R->R means that the function takes real numbers and provides real numer. For ex. If f(x) =2x this means if you give to your function a real number like 2,it gives you also a real number 2times2=4
Adri
pls answer this question *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
If you have f:R^n->R^n you give to your function a vector of length n like (a1,a2,...an) where all a1,.. an are reals and gives you also a vector of length n... I don't know if i answering your question. Otherwise on YouTube you havr many videos where they explain it in a simple way
Adri
I would say 24
Adri
Offer both
Adri
Sorry 20
Adri
Actually you have 40 - 4 =36 who offer maths or physics or both.
Adri
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
Adri
56-36=20 who give both courses... I would say that
Adri
solution: In a question involving sets and Venn diagram, the sum of the members of set A + set B - the joint members of both set A and B + the members that are not in sets A or B = the total members of the set. In symbolic form n(A U B) = n(A) + n (B) - n (A and B) + n (A U B)'.
Mckenzie
In the case of sets A and B use the letters m and p to represent the sets and we have: n (M U P) = 40; n (M) = 24; n (P) = 32; n (M and P) = unknown; n (M U P)' = 4
Mckenzie
Now substitute the numerical values for the symbolic representation 40 = 24 + 32 - n(M and P) + 4 Now solve for the unknown using algebra: 40 = 24 + 32+ 4 - n(M and P) 40 = 60 - n(M and P) Add n(M and P), as well, subtract 40 from both sides of the equation to find the answer.
Mckenzie
40 - 40 + n(M and P) = 60 - 40 - n(M and P) + n(M and P) Solution: n(M and P) = 20
Mckenzie
thanks
Inembo
Simpler form: Add the sums of set M, set P and the complement of the union of sets M and P then subtract the number of students from the total.
Mckenzie
n(M and P) = (32 + 24 + 4) - 40 = 60 - 40 = 20
Mckenzie

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