How to find slope of curve at certain points

how to find slope at certain points circled in blue in below curve ? Are these below 2 approaches valid ? though they give different results . How to automatically find the points where the slope changes drastically in curve like around at point 5,6 in below graph

x=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ]

y=[512, 256, 128, 64 , 32 , 16 , 8  , 7  , 6  , 5  , 4  , 3  , 2  , 1  ]

1. Numpy gradient give below result

[-256. , -192. ,  -96. ,  -48. ,  -24. ,  -12. ,   -4.5,   -1. ,-1. ,   -1. ,   -1. ,   -1. ,   -1. ,   -1. ]


can we use numpy.gradient to find the slope of curve ? since finding slope of line and curve is bit different Shown in this link

2.Using custom slope function

def slope(x1, y1, x2, y2):
m = (y2-y1)/(x2-x1)
return m

slope_value=[]
for i in range(len(y)):
i += 1
v=slope(y[i], x[i], y[i-1], x[i-1])
print(i,v)
slope_value.append(v)

result: [-0.00390625,  -0.0078125,  -0.015625,  -0.03125,  -0.0625,  -0.125,  -1.0,  -1.0,  -1.0,  -1.0,  -1.0,  -1.0,  -1.0]


The numpy calculation is the correct one to use, but may be a bit tricky to understand how it is calculated

Your custom calculation is accidentally returning the inverse slope, the x and y values are reversed in the slope function (x1 -> y[i], etc). The slope should be delta_y/delta_x

def slope(x1, y1, x2, y2):
v=slope(y[i], x[i], y[i-1], x[i-1])


Also, you are calculating the slope at x = 1.5, 2.5, etc but numpy is calculating the slope at x = 1, 2, 3

In the gradient calculation, numpy is calculating the gradient at each x value, by using the x-1 and x+1 values and dividing by the difference in x which is 2. You are calculating the inverse of the x + .5 values.

x=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ]

y=[512, 256, 128, 64 , 32 , 16 , 8  , 7  , 6  , 5  , 4  , 3  , 2  , 1  ]


[-256. , -192. ,  -96. ,  -48. ,  -24. ,  -12. ,   -4.5,   -1. ,-1. ,   -1. ,   -1. ,   -1. ,   -1. ,   -1. ]