Tangent Line to a Graph

Tangent Line to a Graph

How to find the tangent line to a given graph: formula, examples, and their solutions.

Formula

Tangent line to a graph on (a, f(a)): y = f'(a)(x - a) + f(a)

The tangent line to y = f(x) at x = a is
y = f'(a)(x - a) + f(a).

The slope of the tangent line is f'(a).
And the tangent line is on (a, f(a)).

Then, the tangent line in point-slope form is
y = f'(a)(x - a) + f(a).

Example 1

Find the equation of the tangent line to y = f(x) at x = 1. f(x) = x^3 - 4x^2 + 7

Put 1 into f(x).
Then f(1) = 4.

So the tangent line is on (1, 4).

Find the slope of the tangent line.

f'(x) = 3x2 - 8x

Derivatives of polynomials

So f'(1) = -5.

f'(1) = -5
(1, 4)

Then the tangent line is y = -5(x - 1) + 4.

Point-slope form

Example 2

Find the equation of the tangent line to y = f(x) at x = 0. f(x) = e^x

Put 0 into f(x).
Then f(0) = 1.

So the tangent line is on (0, 1).

Find the slope of the tangent line.

f'(x) = ex

Derivative of ex

So f'(0) = 1.

f'(0) = 1
(0, 1)

Then the tangent line is y = 1(x - 0) + 1.

Point-slope form