Write a limit representing the derivative of f(x) at x=1.

Question

Given f(x)=9x2+4x−9, write a limit representing the derivative of f(x) at x=1.

Answer 1

Write a limit representing the derivative of f(x) at x = 1. Given f(x) = 9x² + 4x − 9, write a limit representing the derivative of f(x) at x = 1: Slope at x = 1: m = f ^' (1) = ??

Detailed Solution:

Given: f(x) = 9x² + 4x − 9 ==> f ^' (x) = 18x + 4 (Derivative of f(x))

Slope at x = 1: m = f ^' (1) = 18(1) + 4 = 22 ==> m = 22 (this is the final answer)

However, the question is to arrive to this final answer using the definition of derivative; here it is:

Definition of Derivative: Lim _h=>0 [ f( x + h) – f(x) ]/(h)

Given: f(x) = 9x² + 4x − 9

f ^' (x) = Lim _h=>0 [ f( x + h) – f(x) ]/(h) ==>

f ^' (x) = Lim _h=>0 [ ( 9(x + h)² + 4(x + h) − 9) – (9x² + 4x − 9) ]/(h) ==>

f ^' (x) = Lim _h=>0 [ ( 9(x² + 2xh + h²) + 4(x + h) − 9) – (9x² + 4x − 9) ]/(h) ==>

f ^' (x) = Lim _h=>0 [ 9x² + 18xh + 9h² + 4x + 4h − 9 – 9x² – 4x + 9 ]/(h) ==>

f ^' (x) = Lim _h=>0 [ 18xh + 9h² + 4h ]/(h) ==> f ^' (x) = Lim _h=>0 (h)[ 18x + 9h + 4 ]/(h) ==>

f ^' (x) = Lim _h=>0 [ 18x + 9h + 4 ] ==> f ^' (x) = [ 18x + 9(0) + 4 ] ==> f ^' (x) = 18x + 4 (Derivative of f(x))

Slope at x = 1: m = f ^' (1) = 18(1) + 4 = 22 ==> m = 22 (this is the final answer)

Answer 2

The limit definition of a derivative is:

so just substitute x = 1 and f(x) = 9x² + 4x - 9:

f(1 + h) = 9(1 + h)² + 4(1 + h) - 9 = 9(1 + 2h + h²) + 4 + 4h - 9 = 9 + 18h + 9h² + 4 + 4h - 9 = 9h² + 22h + 4

f(1) = 9(1)² + 4(1) - 9 = 4

so f(1 + h) - f(1) = 9h² + 22h meaning the limit that represents the derivative is: