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7  Finding the equation of a tangent line
Differentiate then use the derivative and a given 'x' value or a given point, to find the gradient of the tangent line y = mx + c Substitute in a known 'x' and 'y' value and solve to find 'c' This will determine the equation of the tangent line.

Ex15.03, Nulake IAS p84,85

Find the gradient of the tangent line (m) (differentiate and substitute in the x value)y =mx + cSubstitute the x & y values of the point into the equation to find c 
Find the equation of the tangent to the curve y = x^{2}  3x + 5 at the point (1,3) 

Steps: 
y = x^{2}  3x + 5 
1) General tangent equation y = mx + c 

2) Differentiate to find gradient m 
y' = 2x  3 
3) Substitute x = 1 into the derivative 
y' = 2x1  3 
4) The result is the gradient m 
m = 1 
Tangent equation so far is y = 1x + c 

5) Substitute in x=1 & y=3 values in to find c 
y = 1x + c 
6) Tangent equation is y = 1x + 4 

Mouse over boxes right 

Find the equation of the normal to the curve y = x^{2}  3x + 5 at the point (1,3) 

Steps: 
y = x^{2}  3x + 5 
1) General normal equation y = mx + c 

2) Differentiate to find gradient of tangent m_{1} 
y' = 2x  3 
3) Substitute x = 1 into the derivative 
y' = 2x1  3 = 1 
4) The result is the gradient of the tangent m_{1} 
m_{1} = 1 
5) Find the negative inverse of m_{1}to give the gradient of the normal line m_{2} 
m_{2}= 1 
Normal equation so far is y = 1x + c 

6) Substitute in x=1 & y=3 values in to find c 
y = 1x + c 
7) Normal equation is y = 1x + 2 
