Suppose that the function f(x) is differentiable.Explain how you would find the equation of the line tangent to the graph of a function f(x) at x=c .Is the tangent line the same as f ′(c)?

Question

Suppose that the function f(x) is differentiable. Explain how you would find the equation of the line tangent to the graph of a function f(x) at x=c. Is the tangent line the same as f′(c)? If not, explain how they are related.

Answer 1

Hi, Ashton,

You're on the right track! f'(c) is the SLOPE of the tangent line. To find the equation of the tangent line, you need 2 things: the SLOPE of the tangent line (which is f'(c)) and a point on the line (which would be the point (c, f(c)))

Given f(x) and a value for x (let's call this 'c'), follow these steps:

STEP 1: Find the derivative, f'(x).

STEP 2: Find the slope of the tangent line by plugging c into f'(x) to find f'(c).

STEP 3: Find a coordinate point that is both on the tangent line and on the original function by plugging c into f(x). (This is the only point on the original function where the tangent line touches it.)

STEP 4: Use the point-slope formula for a line and your answers from Step 2 (f'(c)) & Step 3 (c,f(c)) to find the equation of the tangent line.

y - y₁ = m (x - x₁)

y - f(c) = f'(c) (x - c)

OPTIONAL STEP: Depending on teacher's requirements, rearrange tangent line equation to be in standard y= form.