Question

How do you find the equation of a line tangent to the function `y=4/sqrtx` at (1,4)?

Answer 1

#y=-2x+6

Explanation:

The equation of the tangent in `color(blue)"point-slope form"` is.

`color(red)(bar(ul(|color(white)(2/2)color(black)(y-y_1=m(x-x_1))color(white)(2/2)|)))`
where m represents the slope and `(x_1,y_1)" a point on the line"`

`color(orange)"Reminder"color(white)(xxx)m=dy/dx`

Express `y=4/sqrtx=4/x^(1/2)=4x^(-1/2)`

`rArrdy/dx=-2x^(-3/2)=-2/x^(3/2)`

`x=1:dy/dx=-2/1^(3/2)=-2/1=-2=m`

substitute m = - 2 and `(x_1,y_1)=(1,4)` into the equation.

`y-4=-2(x-1)rArry-4=-2x+2`

`rArry=-2x+6" is the equation of the tangent"`