# 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.

The f'(4)for f(x) =4^x
I need help under implicit differentiation
how to understand this
nhie
understand what?
Boniface
how dont you understand it is obvious
Se
if it's there, it must be differentiable
Giorgio
marginal rate of substitution in economics for an example is an implicit differentiation function. It is a proportion of comparison. It could be expressed as the area of a triangle of the old value proportional to the new, and then the next value and so on.
James
The new shapes form a line with a derivative curve
James
That curve could be expressed mathematically. A good real life example is the proportion at which people barter in pawn shops. "How about 100?" "What are you trying to do? rob me? 50!" "No way chap. 75." "Ill give you 62. deal?" "68. Nothing more nothing less." "deal"
James
the proportion of what someone was trying to get for their product, versus what they were offered to the new price they wanted for their product and what they were offered
James
The proportion of the differentiating triangles would be somewhat 1:2. and since there is little variation to the curve then it looks more like a straight line.
James
By the way This is one of the hardest subjects for me. I have a really hard time expressing things in such a way. I'm trying to have more exact calculations which is why I still study the subject.
James
What is derivative of antilog x dx ?
what's the meaning of removable discontinuity
what's continuous
Brian
an area under a curve is continuous because you are looking at an area that covers a range of numbers, it is over an interval, such as 0 to 4
Lauren
using product rule x^3,x^5
Kabiru
may god be with you
Sunny
Luke 17:21 nor will they say, See here or See there For indeed, the kingdom of God is within you. You've never 'touched' anything. The e-energy field created by your body has pushed other electricfields. even our religions tell us we're the gods. We live in energies connecting us all. Doa/higgsfield
Scott
if you have any calculus questions many of us would be happy to help and you can always learn or even invent your own theories and proofs. math is the laws of logic and reality. its rules are permanent and absolute. you can absolutely learn calculus and through it better understand our existence.
Scott
ya doubtless
Bilal
help the integral of x^2/lnxdx
Levis
also find the value of "X" from the equation that follow (x-1/x)^4 +4(x^2-1/x^2) -6=0 please guy help
Levis
Use integration by parts. Let u=lnx and dv=x2dx Then du=1xdx and v=13x3. ∫x2lnxdx=13x3lnx−∫(13x3⋅1x)dx ∫x2lnxdx=13x3lnx−∫13x2dx ∫x2lnxdx=13x3lnx−19x3+C
Bilal
itz 1/3 and 1/9
Bilal
now you can find the value of X from the above equation easily
Bilal
Pls i need more explanation on this calculus
usman
usman from where do you need help?
Levis
thanks Bilal
Levis
integrate e^cosx
Uchenna
-sinx e^x
Leo
Do we ask only math question? or ANY of the question?
yh
Gbesemete
How do i differentiate between substitution method, partial fraction and algebraic function in integration?
usman
you just have to recognize the problem. there can be multiple ways to solve 1 problem. that's the hardest part about integration
Lauren
test
we asking the question cause only the question will tell us the right answer
Sunny
find integral of sin8xcos12xdx
don't share these childish questions
Bilal
well find the integral of x^x
Levis
bilal kumhar you are so biased if you are an expert what are you doing here lol😎😎😂😂 we are here to learn and beside there are many questions on this chat which you didn't attempt we are helping each other stop being naive and arrogance so give me the integral of x^x
Levis
Levis I am sorry
Bilal
Bilal it okay buddy honestly i am pleasured to meet you
Levis
x^x ... no anti derivative for this function... but we can find definte integral numerically.
Bilal
thank you Bilal Kumhar then how we may find definite integral let say x^x,3,5?
Levis
evaluate 5-×square divided by x+2 find x as limit approaches infinity
i have not understood
Leo
Michael
welcome
Sunny
I just dont get it at all...not understanding
Michagaye
0 baby
Sunny
The denominator is the aggressive one
Sunny
wouldn't be any prime number for x instead ?
Harold
or should I say any prime number greater then 11 ?
Harold
just wondering
Harold
I think as limit Approach infinity then X=0
Levis
ha hakdog hahhahahaha
ha hamburger
Leonito
Fond the value of the six trigonometric function of an angle theta, which terminal side passes through the points(2x½-y)²,4
What's f(x) ^x^x
What's F(x) =x^x^x
Emeka
are you asking for the derivative
Leo
that's means more power for all points
rd
Levis
iam sorry f(x)=x^x it means the output(range ) depends to input(domain) value of x by the power of x that is to say if x=2 then x^x would be 2^2=4 f(x) is the product of X to the power of X its derivatives is found by using product rule y=x^x introduce ln each side we have lny=lnx^x =lny=xlnx
Levis
the derivatives of f(x)=x^x IS (1+lnx)*x^x
Levis
So in that case what will be the answer?
Alice
nice explanation Levis, appreciated..
Thato
what is a maximax
A maxima in a curve refers to the maximum point said curve. The maxima is a point where the gradient of the curve is equal to 0 (dy/dx = 0) and its second derivative value is a negative (d²y/dx² = -ve).
Viewer
what is the limit of x^2+x when x approaches 0
it is 0 because 0 squared Is 0
Leo
0+0=0
Leo
simply put the value of 0 in places of x.....
Tonu
the limit is 2x + 1
Nicholas
the limit is 0
Muzamil
limit s x
Bilal
The limit is 3
Levis
Leo we don't just do like that buddy!!! use first principle y+∆y=x+∆x ∆y=x+∆x-y ∆y=(x+∆x)^2+(x+∆x)-x^2+x on solving it become ∆y=3∆x+∆x^2 as ∆x_>0 limit=3 if you do by calculator say plugging any value of x=0.000005 which approach 0 you get 3
Levis
find derivatives 3√x²+√3x²
3 + 3=6
mujahid
How to do basic integrals
the formula is simple x^n+1/n+1 where n IS NOT EQUAL TO 1 And n stands for power eg integral of x^2 x^2+1/2+1 =X^3/3
Levis