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Tangent Line Formula


A tangent line of any curve is a line that touches the curve at a point called the point of tangency. For a curve y = f(x), the tangent line at a point x = a on the curve is the line that passes through the point(a, f(a)) on the curve and also has a slope equal to f'(a).

The Tangent Line Formula of the curve at any point 'a' is given as,

Tangent Line Formula

Where,
f(a) is the value of the curve function at a point 'a'
m is the value of the derivative of the curve function at a point 'a'

Related Calculators
Equation of Tangent Line Calculator Calculator Tangent
Inverse Tangent Calculator with Sine Cosine and Tangent
 

Tangent Line Problems

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Some solved problems on tangent line are given below:

Solved Examples

Question 1: Find the tangent line of the curve f(x) = 3x2 - 3 at x0 = 0 ?
Solution:
 
Given:
f(x) = 3x2 - 3
x= 0

f(x0) = f(0) = 3(0)2 - 3 = -3
f'(x) = 6x
m = f'(x0) = 6(0) = 0

The tangent line formula is,
y - f(x0) = m(x - x0)
y + 3 = 0(x - 0)
y + 3 = 0

The tangent of the curve is,
y + 3 = 0 
 

Question 2: Find the tangent line of the curve f(x) = 5x2 + 17 at x0 = 2 ?
Solution:
 
Given:
f(x) = 5x2 + 17
x0 = 2

f(x0) = f(2) = 5(2)2 + 17 = 37
f'(x) = 10x
m = f'(x0) = 10(2) = 20

The tangent line formula is,
y - f(x0) = m(x - x0)
y - 37 = 20(x - 2)
y - 37 = 20x - 40
20x - y - 3 = 0

The tangent of the curve is,
20x - y - 3 = 0

 

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