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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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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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