9.6 Gradients of Tangents, Equations of Tangents and Normals
![](http://spmaddmaths.blog.onlinetuition.com.my/wp-content/uploads/sites/3/2013/12/Picture15.png)
![](http://spmaddmaths.blog.onlinetuition.com.my/wp-content/uploads/sites/3/2013/12/Picture15.png)
If A(x1, y1) is a point on a line y = f(x), the gradient of the line (for a straight line) or the gradient of the tangent of the line (for a curve) is the value of
when x = x1.
(A) Gradient of tangent at A(x1, y1):
![](http://spmaddmaths.blog.onlinetuition.com.my/wp-content/uploads/sites/3/2013/12/Picture16.png)
![](http://spmaddmaths.blog.onlinetuition.com.my/wp-content/uploads/sites/3/2013/12/Picture16.png)
(B) Equation of tangent:
![](http://spmaddmaths.blog.onlinetuition.com.my/wp-content/uploads/sites/3/2013/12/Picture17.png)
![](http://spmaddmaths.blog.onlinetuition.com.my/wp-content/uploads/sites/3/2013/12/Picture17.png)
(C) Gradient of normal at A(x1, y1):
![](http://spmaddmaths.blog.onlinetuition.com.my/wp-content/uploads/sites/3/2013/12/Picture20.png)
![](http://spmaddmaths.blog.onlinetuition.com.my/wp-content/uploads/sites/3/2013/12/Picture18.png)
![](http://spmaddmaths.blog.onlinetuition.com.my/wp-content/uploads/sites/3/2013/12/Picture20.png)
![](http://spmaddmaths.blog.onlinetuition.com.my/wp-content/uploads/sites/3/2013/12/Picture18.png)
(D) Equation of normal :
![](http://spmaddmaths.blog.onlinetuition.com.my/wp-content/uploads/sites/3/2013/12/Picture19.png)
![](http://spmaddmaths.blog.onlinetuition.com.my/wp-content/uploads/sites/3/2013/12/Picture19.png)
Example 1 (Find the Equation of Tangent)
Given that
. Find the equation of the tangent at the point (1, 1).
Solution:
Example 2 (Find the Equation of Normal)
Solution:
Find the gradient of the curve
at the point (-1, 7). Hence, find the equation of the normal to the curve at this point.
Solution: