University:
University of Cambridge
Course:
0580 IGCSE | Mathematics
Academic year:

2023
Views:

351

Pages:

3
Author:

Shahbaz A.

Implicit Differentiation and equation of a tangent line through a point shahbaz ahmed August 2024 Differentiating Implicitly, find the equation of tangent line through the point (5,10) xm y n = (x + y)m+n Solution xm y n = (x + y)m+n ln xm y n = = ln (x + y)m+n ln xm + ln y n = (m + n) ln (x + y) m ln x + n ln y = (m + n) ln (x + y) 1 d d [m ln x + n ln y] = [(m + n) ln (x + y)] dx dx d d d [m ln x] + [n ln y] = (m + n) [ln (x + y)] dx dx dx m d d 1 d [ln x] + n [ln y] = (m + n) [(x + y)] dx dx x + y dx m 1 dx 1 dy 1 dx dy +n = (m + n) [ + ] x dx y dx x + y dx dx m n dy m+n dy + = [1 + ] x y dx x+y dx n dy m + n dy m+n m − =[ − ] y dx x + y dx x+y x [ n m + n dy m+n m − ] =[ − ] y x + y dx x+y x [ n(x + y) − (m + n)y dy (m + n)x − m(x + y) ] =[ ] y(x + y) dx x(x + y [ nx + ny − my − ny dy mx + nx − mx − my) ] =[ ] y(x + y) dx x(x + y 2 [ nx − my dy nx − my) ] =[ ] y(x + y) dx x(x + y nx − my) y(x + y) dy =[ ] × dx x(x + y nx − my dy y = dx x Putting y=10,x=5, the slope m of the tangent line m= dy dx = 10 5 =2 Equation of the line through the point (5,10) with slope m=2 y-10=2(x-5) y-10=2x-10 2x-10=y-10 2x-10-y+10=0 2x-y=0 3