HOW TO FIND THE SLOPE OF A TANGENT LINE AT A POINT

In this section, we are going to see how to find the slope of a tangent line at a point.

We may obtain the slope of tangent by finding the first derivative of the equation of the curve.

If y  =  f(x) is the equation of the curve, then f'(x) will be its slope.

So, slope of the tangent is 

m  =  f'(x)  or  dy/dx

Let us look into some examples to understand the above concept.

Example 1 :

Find the equation of the slope of tangent to the parabola y² = 12x at the point (3, 6)

Solution :

Equation of the given curve is y² = 12x

2y (dy/dx)  =   12 (1)

2y (dy/dx)  =   12

dy/dx  =  12/2y  ==>  6/y

Slope of tangent at (3, 6) is

  m  =  6/6

  m  =  1

Hence the slope of the tangent line at the given point is 1.

Example 2 :

Find the equation of the tangent to the parabola x² + x − 2y + 2 = 0 at (1, 2)

Solution :

Equation of the given curve x² + x − 2y + 2 = 0

2x + 1 - 2 (dy/dx) + 0  =  0

2 (dy/dx)  =  2x + 1

dy/dx  =  (2x + 1)/2

Slope of tangent at (1, 2) :

dy/dx  =  (2(1) + 1)/2  =  3/2

Hence the slope of tangent at the given point (1, 2) is 3/2.

Example 3 :

Find the equation of the tangent to the hyperbola 9x²- 5y² = 31 at (2, -1)

Solution :

Equation of the given curve is 9x²- 5y² = 31 

18 x - 10 y (dy/dx)  =  0

10y (dy/dx)  =  -18 x

dy/dx  =  -18x / 10  ==>  -9x/5

Slope of tangent at the point (2, -1)

m  = -9(2)/5  ==>  -18/5

So, the slope of the tangent at the given point is -18/5.

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