Question

What is the equation of the tangent line of `f(x) =sqrt(x+15)-x/(x-15)` at `x=5`?

Answer 1

`20y-(3+sqrt5)x+5-35sqrt5=0`

Explanation:

Tangent Line to a Curve

`y=sqrt(x+15)-x/(x-15)`

Substitute for `x` value to get the coordinates of the point which lies on the tangent

`y=(1+4sqrt5)/2`

Differentiate the function Using

Power rule ,Quotient rule , Chain rule

`y'=1/(2sqrt(x+15))-((x-15)xx1-x xx1)/(x-15)^2`

Simplify

`y'=1/(2sqrt(x+15))+15/(x-15)^2`

Now We substitute for x=5 to get the Slope of the Tangent `m`

`y'_(x=5)=(3+sqrt5)/20`=`m`

now Substitute in the straight line equation

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

`y-(1+4sqrt5)/2=(3+sqrt5)/20(x-5)`

`20y-(3+sqrt5)x+5-35sqrt5=0`