Calculus Derivatives

Question

Find an equation for the line tangent to the graph of

f(x) = (2x + 3) / (3x - 2)

at the point P(1, f(1))

a. -2x + 3y = 13

b. x + 13y = 66

c. 13x + y = 18

d. 13x - y = 8

Answer 1

We want to find the line tangent to f(x) = (2x+3)/(3x-2) at the point (1,f(1)).

First, let's find f(1).

f(1)=(2(1)+3) / (3(1)-2)

f(1)=(5)/(1)=5

So, we are looking for the tangent at (1,5).

Let's find the slope at x=1. To do this we're going to take the derivative of f(x), and then take f '(1).

Using the quotient rule we find that the derivative is f '(x)=−13/(3x−2)².

f '(1)=-13/(3(1)-2)²

f '(1)= -13/(1)=-13

So now that we have the derivative at f(1), we just use point slope form to find the equation of the tangent line, which I'll call g(x).

g(x)-5=(-13)(x-1)

g(x)= -13x+13+5

g(x)= -13x +18

Let me know if you need another example!

-Ethan