How to find the slope of a tangent line at a point :
Here 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 curve.
If f(x) is the equation of the curve, then f'(x) will be its slope.
f'(x) = dy/dx = m (slope of tangent)
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)
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)
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)
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
Hence the slope of the tangent at the given point is -18/5.
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