2. Given the equation x^2 - 5, find the derivative of the function using first principles and then find the tangent.

Question

Two part question:

Given the equation x^2 - 5,

a. Find the derivative of the function using first principles

b. find the tangent

Answer 1

First principles means use the definition of the derivative:

f(x) = x^2 - 5

df(x)/dx = lim as h --> 0 (x+h)^2 - 5 - (x^2-5)/h

= lim as h --> 0 (x^2 + 2hx + h^2 - 5 - x^2 +5)/h

= lim as h --> 0 (2hx + h^2)/h

= lim as h --> 0 (2x + h)

= 2x

The derivative is the slope of the tangent line at every value of x. So at (3,4) the slope of the tangent line is 2x = 2(3) = 6. The equation of a line is y = mx + b where m is the slope and b is the y-intercept. For the tangent line so far we have y = 6x + b. To find the value of b for the tangent line, plug the given point (3,4) into y = 6x + b and solve for b.