Question

For `f(x) =(5+6x)^2`, what is the equation of the line tangent to `x =4`?

Answer 1

y - 348x + 551 = 0

Explanation:

y - b = m(x - a) is the equation of the tangent. Require to

find m (slope ) and (a , b ) , a point on the tangent.

[ m = f'(x) ,a is x = 4 and b can be found by substituting x =4

into f(x). ]

differentiate f(x) using `color(blue)(" chain rule")`

hence f'(x) = `2(5 + 6x ) d/dx (5 + 6x )`

`= 2(5 + 6x ).6 = 12 (5 + 6x )`

now m = f'(4) = 12( 5 + 24 ) = 348

and f(4) = (5 + 24 )^2 = 841 → (a , b ) = ( 4 , 841 )

equation is : y - 841 = 348 (x - 4 )

so y - 841 = 348x - 1392

hence: y - 348x - 551 = 0