Equation of a Tangent Line (Derivative)
See how to find the equation of a tangent line
by using the derivative of a function.
2 examples and their solutions.
Equation of a Tangent Line (Derivative)
Formula
y = f'(a)(x - a) + f(a)
Derivative Rules
Tangent line passes through (a, f(a)).
→ y - f(a) = f'(a)(x - a)
Linear Equation (Two Variables)
→ y = f'(a)(x - a) + f(a)
Example
f(x) = x3 - 4x2 + 7
Tangent line at x = 1?
Solution Tangent line at x = 1?
f(x) = x3 - 4x2 + 7
f(1) = 13 - 4⋅12 + 7
= 1 - 4⋅1 + 7
= 1 - 4 + 7
= 4
→ (1, 4)
f'(x) = 3x2 - 4⋅2x1 + 0 - [1]
= 3x2 - 8x
f'(1) = 3⋅12 - 8⋅1
= 3⋅1 - 8
= 3 - 8
= -5
→ y = -5(x - 1) + 4
= -5x + 5 + 4
y = -5x + 9
f(1) = 13 - 4⋅12 + 7
= 1 - 4⋅1 + 7
= 1 - 4 + 7
= 4
→ (1, 4)
f'(x) = 3x2 - 4⋅2x1 + 0 - [1]
= 3x2 - 8x
f'(1) = 3⋅12 - 8⋅1
= 3⋅1 - 8
= 3 - 8
= -5
→ y = -5(x - 1) + 4
= -5x + 5 + 4
y = -5x + 9
Close
Example
f(x) = ex
Tangent line at x = 0?
Solution Tangent line at x = 0?
f(x) = ex
f(0) = e0
= 1 - [1]
→ (0, 1)
f'(x) = ex - [2]
f'(0) = e0
= 1
→ y = 1(x - 0) + 1
y = x + 1
f(0) = e0
= 1 - [1]
→ (0, 1)
f'(x) = ex - [2]
f'(0) = e0
= 1
→ y = 1(x - 0) + 1
y = x + 1
Close