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  • Define the derivative function of a given function.
  • Graph a derivative function from the graph of a given function.
  • State the connection between derivatives and continuity.
  • Describe three conditions for when a function does not have a derivative.
  • Explain the meaning of a higher-order derivative.

As we have seen, the derivative of a function at a given point gives us the rate of change or slope of the tangent line to the function at that point. If we differentiate a position function at a given time, we obtain the velocity at that time. It seems reasonable to conclude that knowing the derivative of the function at every point would produce valuable information about the behavior of the function. However, the process of finding the derivative at even a handful of values using the techniques of the preceding section would quickly become quite tedious. In this section we define the derivative function and learn a process for finding it.

Derivative functions

The derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined. We can formally define a derivative function as follows.


Let f be a function. The derivative function    , denoted by f , is the function whose domain consists of those values of x such that the following limit exists:

f ( x ) = lim h 0 f ( x + h ) f ( x ) h .

A function f ( x ) is said to be differentiable at a    if f ( a ) exists. More generally, a function is said to be differentiable on S    if it is differentiable at every point in an open set S , and a differentiable function    is one in which f ( x ) exists on its domain.

In the next few examples we use [link] to find the derivative of a function.

Finding the derivative of a square-root function

Find the derivative of f ( x ) = x .

Start directly with the definition of the derivative function. Use [link] .

f ( x ) = lim h 0 x + h x h Substitute f ( x + h ) = x + h and f ( x ) = x into f ( x ) = lim h 0 f ( x + h ) f ( x ) h . = lim h 0 x + h x h · x + h + x x + h + x Multiply numerator and denominator by x + h + x without distributing in the denominator. = lim h 0 h h ( x + h + x ) Multiply the numerators and simplify. = lim h 0 1 ( x + h + x ) Cancel the h . = 1 2 x Evaluate the limit.
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Finding the derivative of a quadratic function

Find the derivative of the function f ( x ) = x 2 2 x .

Follow the same procedure here, but without having to multiply by the conjugate.

f ( x ) = lim h 0 ( ( x + h ) 2 2 ( x + h ) ) ( x 2 2 x ) h Substitute f ( x + h ) = ( x + h ) 2 2 ( x + h ) and f ( x ) = x 2 2 x into f ( x ) = lim h 0 f ( x + h ) f ( x ) h . = lim h 0 x 2 + 2 x h + h 2 2 x 2 h x 2 + 2 x h Expand ( x + h ) 2 2 ( x + h ) . = lim h 0 2 x h 2 h + h 2 h Simplify. = lim h 0 h ( 2 x 2 + h ) h Factor out h from the numerator. = lim h 0 ( 2 x 2 + h ) Cancel the common factor of h . = 2 x 2 Evaluate the limit.
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Find the derivative of f ( x ) = x 2 .

f ( x ) = 2 x

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We use a variety of different notations to express the derivative of a function. In [link] we showed that if f ( x ) = x 2 2 x , then f ( x ) = 2 x 2 . If we had expressed this function in the form y = x 2 2 x , we could have expressed the derivative as y = 2 x 2 or d y d x = 2 x 2 . We could have conveyed the same information by writing d d x ( x 2 2 x ) = 2 x 2 . Thus, for the function y = f ( x ) , each of the following notations represents the derivative of f ( x ) :

f ( x ) , d y d x , y , d d x ( f ( x ) ) .

Questions & Answers

What is derivative of antilog x dx ?
Tanmay Reply
what's the meaning of removable discontinuity
Brian Reply
what's continuous
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
using product rule x^3,x^5
please help me to calculus
World Reply
may god be with you
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
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.
ya doubtless
help the integral of x^2/lnxdx
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
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
itz 1/3 and 1/9
now you can find the value of X from the above equation easily
Pls i need more explanation on this calculus
usman from where do you need help?
thanks Bilal
Do we ask only math question? or ANY of the question?
Levis Reply
How do i differentiate between substitution method, partial fraction and algebraic function in integration?
you just have to recognize the problem. there can be multiple ways to solve 1 problem. that's the hardest part about integration
we asking the question cause only the question will tell us the right answer
find integral of sin8xcos12xdx
Levis Reply
don't share these childish questions
well find the integral of x^x
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 I am sorry
Bilal it okay buddy honestly i am pleasured to meet you
x^x ... no anti derivative for this function... but we can find definte integral numerically.
thank you Bilal Kumhar then how we may find definite integral let say x^x,3,5?
evaluate 5-×square divided by x+2 find x as limit approaches infinity
Michagaye Reply
i have not understood
The answer is 0
I just dont get it at all...not understanding
0 baby
The denominator is the aggressive one
wouldn't be any prime number for x instead ?
or should I say any prime number greater then 11 ?
just wondering
I think as limit Approach infinity then X=0
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No Reply
ha hamburger
Fond the value of the six trigonometric function of an angle theta, which terminal side passes through the points(2x½-y)²,4
albert Reply
What's f(x) ^x^x
Emeka Reply
What's F(x) =x^x^x
are you asking for the derivative
that's means more power for all points
if your asking for derivative dy/dz=x^2/2(lnx-1/2)
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
the derivatives of f(x)=x^x IS (1+lnx)*x^x
what is a maximax
Chinye Reply
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).
what is the limit of x^2+x when x approaches 0
Dike Reply
it is 0 because 0 squared Is 0
simply put the value of 0 in places of x.....
the limit is 2x + 1
the limit is 0
limit s x
The limit is 3
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
find derivatives 3√x²+√3x²
Care Reply
3 + 3=6
How to do basic integrals
dondi Reply
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
write something lmit
ram Reply
find the integral of tan tanxdx
Lateef Reply
-ln|cosx| + C
Practice Key Terms 5

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