Iโm confused with partial derivatives. Since for partial derivatives we can treat all other variables as constants, shouldnโt the derivative vector be [6x_1, 5e^x_2] ?
โf/โx_1 = โ/โx_1 (3x_1^2) + DC = 6x_1 + 0 = 6x_1 (C being a constant)
โf/โx_2 = DC + โ/โx_2 (5e^x_2) = 0 + 5e^x_2 = 5e^x_2
Iโm not clear on what the notation implies when there is both a subscript F and a superscript 2. The text reads as if the Frobenius Norm is always the square root of the sum of its matrix elements, so the superscript should always be 2. Is this understanding incorrect?
def f(x):
return x ** 3 - 1.0 / x
def df(x):
return 3 * x ** 2 + 1/ (x * x)
def tangentLine(x, x0):
"""x is the input list, x0 is the point we compute the tangent line"""
y0 = f(x0)
a = df(x0)
b = y0 - a * x0
return a * x + b
x = np.arange(0.1, 3, 0.1)
plot(x, [f(x), tangentLine(x, 2.1)], 'x', 'f(x)', legend=['f(x)', 'Tangent line (x=2.1)'])
The function is not in the form of any rules given up to now.
Taking the logarithm on both sides yields
Ex5.
Any point such that indicates a stationary point of a function, where a โballโ standing on the point will hold its position and will not fall along the curve.
The stationary point is necessary for a point to be a minimum or maximum point. Sometimes it indicates the critical behavior of a function.
Example: at
Ex6.
The derivative
takes value 4 at point 1.
The tangent line at point 1 is thus
f = lambda x: x ** 3 - 1 / x
x = np.arange(0, 3, 0.1)
plot(x, [f(x), 4 * x - 4], 'x', 'f(x)', legend=['f(x)', 'Tangent line (x=1)'])
such that 
indicates a stationary point of a function, where a โballโ standing on the point will hold its position and will not fall along the curve.
at 
, the derivative is not well-defined, since 
