definition of the derivative

Question

For the function f(x)=x/(x-3)

Use the definition of the derivative to find the value of f '(1)

Write the equation of the line tangent to the curve at (1,f(1))

Answer 1

You will use the formula:

f'(a) = lim_h->0 (f(a+h) - f(a))/h

f'(1) = lim_h->0 ((1+h)/(1+h-3)-(1)/(1-3))/h

= lim_h->0 ((1+h)/(h-2)+(1/2))/h

= lim_h->0 3/(2h-4)

= -3/4

The slope of the tangent at f(1) is -3/4

f(1) = -1/2 so the tangent goes through the point (1, -1/2)

Plug these values into the formula for a straight line:

y - (-1/2) = -3/4(x - 1)

y = -3/4x + 1/4