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Notations for the derivative

The equation of the derivative of a function f ( x ) is written as y = f ( x ) , where y = f ( x ) . The notation f ( x ) is read as “ f  prime of  x . ” Alternate notations for the derivative include the following:

f ( x ) = y = d y d x = d f d x = d d x f ( x ) = D f ( x )

The expression f ( x ) is now a function of x ; this function gives the slope of the curve y = f ( x ) at any value of x . The derivative of a function f ( x ) at a point x = a is denoted f ( a ) .

Given a function f , find the derivative by applying the definition of the derivative.

  1. Calculate f ( a + h ) .
  2. Calculate f ( a ) .
  3. Substitute and simplify f ( a + h ) f ( a ) h .
  4. Evaluate the limit if it exists: f ( a ) = lim h 0 f ( a + h ) f ( a ) h .

Finding the derivative of a polynomial function

Find the derivative of the function f ( x ) = x 2 3 x + 5 at x = a .

We have:

f ( a ) = lim h 0 f ( a + h ) f ( a ) h                   Definition of a derivative 

Substitute f ( a + h ) = ( a + h ) 2 3 ( a + h ) + 5 and f ( a ) = a 2 3 a + 5.

f ( a ) = lim h 0 ( a + h ) ( a + h ) 3 ( a + h ) + 5 ( a 2 3 a + 5 ) h          = lim h 0 a 2 + 2 a h + h 2 3 a 3 h + 5 a 2 + 3 a 5 h Evaluate to remove parentheses .          = lim h 0 a 2 + 2 a h + h 2 3 a 3 h + 5 a 2 + 3 a 5 h Simplify .          = lim h 0 2 a h + h 2 3 h h          = lim h 0 h ( 2 a + h 3 ) h Factor out an  h .          = 2 a + 0 3 Evaluate the limit .          = 2 a 3
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Find the derivative of the function f ( x ) = 3 x 2 + 7 x at x = a .

f ( a ) = 6 a + 7

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Finding derivatives of rational functions

To find the derivative of a rational function, we will sometimes simplify the expression using algebraic techniques we have already learned.

Finding the derivative of a rational function

Find the derivative of the function f ( x ) = 3 + x 2 x at x = a .

f ( a ) = lim h 0 f ( a + h ) f ( a ) h          = lim h 0 3 + ( a + h ) 2 ( a + h ) ( 3 + a 2 a ) h Substitute  f ( a + h )  and  f ( a )         = lim h 0 ( 2 ( a + h ) ) ( 2 a ) [ 3 + ( a + h ) 2 ( a + h ) ( 3 + a 2 a ) ] ( 2 ( a + h ) ) ( 2 a ) ( h ) Multiply numerator and denominator by  ( 2 ( a + h ) ) ( 2 a )         = lim h 0 ( 2 ( a + h ) ) ( 2 a ) ( 3 + ( a + h ) ( 2 ( a + h ) ) ) ( 2 ( a + h ) ) ( 2 a ) ( 3 + a 2 a ) ( 2 ( a + h ) ) ( 2 a ) ( h ) Distribute         = lim h 0 6 3 a + 2 a a 2 + 2 h a h 6 + 3 a + 3 h 2 a + a 2 + a h ( 2 ( a + h ) ) ( 2 a ) ( h ) Multiply         = lim h 0 5 h ( 2 ( a + h ) ) ( 2 a ) ( h ) Combine like terms         = lim h 0 5 ( 2 ( a + h ) ) ( 2 a ) Cancel like factors         = 5 ( 2 ( a + 0 ) ) ( 2 a ) = 5 ( 2 a ) ( 2 a ) = 5 ( 2 a ) 2 Evaluate the limit

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Find the derivative of the function f ( x ) = 10 x + 11 5 x + 4 at x = a .

f ( a ) = 15 ( 5 a + 4 ) 2

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Finding derivatives of functions with roots

To find derivatives of functions with roots, we use the methods we have learned to find limits of functions with roots, including multiplying by a conjugate.

Finding the derivative of a function with a root

Find the derivative    of the function f ( x ) = 4 x at x = 36.

We have

f ( a ) = lim h 0 f ( a + h ) f ( a ) h          = lim h 0 4 a + h 4 a h Substitute  f ( a + h )  and  f ( a )

Multiply the numerator and denominator by the conjugate: 4 a + h + 4 a 4 a + h + 4 a .

    f ( a ) = lim h 0 ( 4 a + h 4 a h ) ( 4 a + h + 4 a 4 a + h + 4 a )             = lim h 0 ( 16 ( a + h ) 16 a h 4 ( a + h + 4 a ) ) Multiply .             = lim h 0 ( 16 a + 16 h 16 a h 4 ( a + h + 4 a ) ) Distribute and combine like terms .             = lim h 0 ( 16 h h ( 4 a + h + 4 a ) ) Simplify .             = lim h 0 ( 16 4 a + h + 4 a ) Evaluate the limit by letting  h = 0.             = 16 8 a = 2 a   f ( 36 ) = 2 36 Evaluate the derivative at  x = 36.             = 2 6             = 1 3
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Find the derivative of the function f ( x ) = 9 x at x = 9.

3 2

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Questions & Answers

Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
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how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
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Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
Practice Key Terms 7

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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