Use the limit definition of the derivative to find the slope of the tangent line to the curve f(x)=4x^2 at x=3

Question

Use the limit definition of the derivative to find the slope of the tangent line to the curve f(x)=4x^2 at x=3

Answer 1

Limit definition of derivative is

 

lim          f(x + h) - f(x)

h-->0     ____________

                       h

 

 

 

lim            4(x + h)² - 4x²

h-->0      ______________

                          h

 

 

 

Now, we can plug in x=3 into the limit definition.

 

 

lim          4(3 + h)² - 4(3)²

h-->0     _______________

                        h

 

 

 

lim         4(3 + h)(3 + h) - 36

h-->0     _________________

                            h

 

 

 

lim         4(9 + 6h + h²) - 36

h-->0     _________________

                            h

 

 

lim             36 + 24h + 4h² - 36

 

h-->0     _____________________

                             h

 

 

 

lim            4h(h + 6)

h-->0      _________

                      h

 

 

lim           4(h + 6)

h-->0

 

 

 

lim           24

h-->0

 

 

The slope of the tangent line at x=3 is 24.