Tangent Line Formula

Introduction To Tangent Line Formula:

Normally Tangent Line is defined as the line which locally just touches a curve particularly at one and only one point .Specifically We have to know that there is no intersection at all in any other points in a curve.Tangent Line Formula is used to represnt the tangent line in an efficient manner.In this article we will see about the tangent line formula with some practice example problems.

Tangent Line Formula:
The tangent line  to a curve at a given point is the straight line that just touches the curve particular point. The point of tangency means that the tangent line is going in the same direction with the curve, and  it is the best straight-line approximation to the curve at that point.

Normally To find the formula for the Tangent line to the curve y=f(x) at point `p(x_o,f(x_0))` ,We need to find out the slope of the curve . Slope of the curve is also called as the gradient of the curve.

Slope of the curve  calculated by using the following two steps:

1.First we need to take the derivative of the curve equation,

`dy/dx=f'(x)`

2. Then evaluate the above equation at the point p(x_o,f(x_0))

`dy/dx=f'(x_0)=slope`

The Equation for the line if we are knowing the slope and point

`y-y_1=m(x-x_1)`

Here the points are`(x_0,f(x_0)` and slope is `f'(x_0)`

So the Tangent line formula is

`y-f(x_0)=f'(x_0)(x-x_0)`

Tangent Line Formula Example1:

Find the equation of the tangent line to the curve y = x4 at the point (2, 8).

Solution:

`dy/dx= 4x^2`

Slope of the Curve when  x = 2 is

4 × 22 = 16.
So apply the slope and points in the slope point formula of the line.

`y-y_1=m(x-x_1)`

`y - 8 = 16(x - 2)`

y = 16x – 24

This is the equation of the tangent line

Tangent Line Formula Example2:

Find the equation of the tangent line to the curve `y = 5x^3-4y^2`  at the point (1, 2).

Solution:

`dy/dx= 15x^2-8y`
Slope of the Curve  when the points are (1,2) is

`dy/dx=15-16`

`m =-1`

So apply the slope value and x and y values into the slope point formula of the line.

`y-y_1=m(x-x_1)`

` y - 2 = -1(x - 1)`

`y-2 = -x +1`

y+x=3

This is the equation of the tangent line