The Chain Rule is the core rule for differentiating composite functions in calculus, described as:
if the function $$ y=f(g(x)) $$
then the derivative of x is:
$$
\frac{dy}{dx} = \frac{dy}{dg}*\frac{dg}{dx}
$$
The code:
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import numpy as np
class Variable:
def __init__(self, data):
self.data = data
self.grad = None
class Function:
def __call__(self, input):
x = input.data
y = self.forward(x)
output = Variable(y)
self.input = input
return output
def forward(self, x):
raise NotImplementedError()
def backward(self, gy):
raise NotImplementedError()
class Square(Function):
def forward(self, x):
y = x ** 2
return y
def backward(self, gy):
x = self.input.data
gx = 2 * x * gy
return gx
class Exp(Function):
def forward(self, x):
y = np.exp(x)
return y
def backward(self, gy):
x = self.input.data
gx = np.exp(x) * gy
return gx
A = Square()
B = Exp()
C = Square()
x = Variable(np.array(0.5))
a = A(x)
b = B(a)
y = C(b)
y.grad = np.array(1.0)
b.grad = C.backward(y.grad)
a.grad = B.backward(b.grad)
x.grad = A.backward(a.grad)
print(x.grad)
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the function is
$$
y = \left(e^{x^{2}}\right)^{2} = e^{2x^{2}}
$$
set
$$
u = 2*x^2
$$
then
$$
y = e^u
$$
so
$$
\frac{dy}{du} = e^{u}
$$
$$
\frac{du}{dx} = 4x
$$
$$
\frac{dy}{dx} = e^{2x^{2}} \cdot 4x
$$
final answer is
$$
\frac{dy}{dx} =\boxed{4x \cdot e^{2x^{2}}}
$$
when x is 0.5
$$
\frac{dy}{dx} = \boxed{4*0.5 \cdot e^{2*(0.5)^{2}}}
$$
same with the code result
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(deep-learning-from-scratch-3) D:\project\deep-learning-from-scratch-3>python steps\step06.py
3.297442541400256
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