Question

How do you find the equation of the line tangent to the graph of `y = e^(-x^2)` at the point (2, 1/e^4)?

Answer 1

1. find derivative of function
2. sub in x value of point to find gradient of tangent
3. put gradient into y=(gradient)x+c
4. Sub in point and solve for c
5. you have found the equation of the tangent.

Explanation:

To find the Equation of a tangent to any curve you must first find the derivative of the function.
For example for `y= e^(-x^2)` => `y'= -2x e^(-x^2)`

sub x=2 into derivative function
`y'(2) = -4e^-4`
so we have found that the tangent line is
`y=(-4e^-4)x + c`

so now we have to sub in the point and solve for c:

`1/e^4=(-4e^-4)(2) + c`
c= `9e^-4`
so the tangent line is equal to
`y=(-4e^-4)x + 9e^-4`
which can be simplified to:
`y=-(4x-9)e^-4`

hope that helped.