# Calculus – Find the Equation of the Tangent Line at (1,3)

By | October 27, 2016

## Find the Equation of the Tangent Line at (1,3)

In this tutorial students learn how to find the equation of the tangent line given the point of tangency. The students then verify the solution using the GRAPH, WINDOW, CALC, dy/dx and TRACE features of the TI-84C.

1) If only the x value is given we should first find the point of tangency by inputting that x value into the function to get the y value.  This (x,y) coordinate is the point of tangency.

2) Calculate the first derivative of the function. This expression represents the slope of the tangent line.

3) Input the x value into the first derivative to get the numerical value of the first derivative (slope).

4a) Use the point slope formula to find the equation of the tangent line.

y-y1=m(x-x1), where the point of tangency is (x1,y1) and the slope is m.

4b) Substitute in the values for x1, y1, and m.

5) Verify your results using the Texas Instrument Graphing Calculator. TI-84, TI-84 Plus, TI-84 C

You have just found the equation of the line tangent to the function at the specified x value.

## Explicit Function

$y=f(x)=2x+\frac{1}{x}$

## Derivative

$y=f(x)=2-\frac{1}{x^2}$

## Point of Tangency

$(1,3)$

## Slope

Substitute in the value of x into the expression of the derivative to find the slope
$y=f(1)=2-\frac{1}{1^2}=-1$

## Tangent Line

$y=x+2$

## Graphing Tool

Click on the link below to see an interactive online graphing calculator based on the equations above.