Equation of tangent line

Question

 Find the equation of the tangent line to the graph of the given function at the point with the indicated x-coordinate.

f(x) = x + (1/x);  x  = 3;   y = ?

 

Answer 1

The curve f(x) =  x + 1/x

y = x + 1/x

Differentiating  on each side with respect of x .

y ' = 1 - 1/x²

Substitute the value x = 3 in above equation.

y ' = 1 - 1/3²

y ' = 1 - 1/9

y ' = (9 - 1)/9

y ' = 8/9

This is the slope of tangent line to the curve at x = 3

m = 8/9

Tangent line touches the function at x = 3.

f(x) =  x + 1/x

f(3) = 3 + 1/3

f(3) = 10/3

Tangent line passes through (3, 10/3).

To find the tangent line equation, substitute the values of m = 8/9 and (x, y) = (3, 10/3).  in the point slope form of an equation y - y₁ = m( x - x₁).

y - 10/3 = 8/9 ( x - 3)

y - 10/3 = 8x/9 - 8/3

y = 8x/9 - 8/3 + 10/3

y = 8x/9  + 2/3