
Applications of the Derivative to Geometry 6-11
And the equation of the tangent at (4, -25) is
y x
x y
+ = − −
⇒ + + =
25 1 4
21 0
( )( )
.
Example 6.11 Find the equation of the tangent to a curve at the point P(2, 3) if the slope of the tangent
to the curve at P is 5.
Solution: We know that the equation of the tangent at any point P(x
1
, y
1
) is
y y
dy
dx
x x
P x y
− =
−
1 1
1 1
( , )
( ),
where
dy
dx
P x y
( , )
1 1
is the slope of the tangent at the point P.
So, we have
y x
x y
− = −
⇒ − − =
3 5 2
5 7 0
( )
.
Therefore, the equation of the tangent is 5x - y - 7 = 0.
Example 6.12 Find the equation of the tangent to the curve e
sin x
+ e
sin y
= 9 at (0, ...