# Tangent Line Equation Calculator

\( \)\( \)\( \)\( \)A step by step tangent line equation calculator is presented.

## Equation of the Tangent Line

Let \( f(x) \) be a function. The slope \( m \) of the tangent line to the graph of \( f(x) \) at the point of tangency \( (x_0 , f(x_0)) \) is given by:

\[ m = f'(x_0) \]

where \( f'(x_0) \) is the first derivative of \( f(x) \) evaluated to \( x = x_0 \)

The equation of the tangent to the graph of \( f(x) \) at \( x = x_0\) in point slope
form is given by

\( y - f(x_0) = m(x - x_0) \)

and in slope intercept
form is given by

\( y = m x + f(x_0) - m x_0 \)

You enter \( f(x) \) and \( x_0 \) and the calculator displays the point of tangency \( ( x_0 , f(x_0) ) \), the slope \( m \) and the equation of the line in slope intercept form \( y = m x + b \) with the y intercept \( b = f(x_0) - m x_0 \).

## Use of the Tangent Line Calculator

1 - Enter and edit function $f(x)$ and click "Enter Function" then check what you have entered. Enter \( x_0 \)

Note that the five operators used are: + (plus) , - (minus), / (division) , ^ (power) and * (multiplication). (example: f(x) = x^3 + 1/x.(more notes on editing functions are located below)

2 - Click "Calculate Equations".

3 - Note that the natural logarirthm is entered as \( log(x) \), the natural exponential as \( exp(x) \).

4 - Note that a function \( f(x) \) to some power \(n\) is entered as: \( (f(x))^n \). Example: \( sin^2(2x-1) \) is entered as (sin(2x-1))^2.

5 - Note Enter decimal numbers as fractions between brackets. Example : enter (1/2) instead of 0.5

Notes: In editing functions, use the following:

1 - The five operators used are: + (plus) , - (minus), / (division) , ^ (power) and * (multiplication). (example: f(x) = x^2 + 1/x + log(x) )

2 - The function square root function is written as (sqrt). (example: sqrt(x^2-1) for \( \sqrt {x^2 - 1} \) )

3 - The exponential function is written as exp(x). (Example: exp(x+2) for \( e^{x+2} \) )

4 - The log base e function is written as log(x). (Example: log(x^2-2) for \( \ln(x^2 - 2 \) )

Here are some examples of functions that you may copy and paste to practice:

x^2 + x +2 sin(x) + cos(x) 1/(x-2) x^2+log(2*x + 2) (x+2)^2(x^2+1)-1

2*sin(2x^2+2x-1) exp(2x^2) tan(x) (x-1)/(x+3)^3

## More References and Links

Tangent Linederivative

rules

formulas