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# 练习 1 错误版本，画成了导数图像
def f_1(x):
return x ** 3.0 - 1.0 / x
x = np.arange(0.1, 2, 0.05) # 根据函数图像特点修改了自变量范围
plot(x, [f_1(x), 3 * x ** 2 + 1 / x ** 2], 'x', 'f(x)', legend=['f(x)', 'Tangent line (x=1)'], xlim=[0.1, 2], ylim=[-2,4])
# 练习 1 正确版本，略过了手算过程，即f_1'(1) = 4 代入(1, 0) 可求出斜截式切线y = 4x - 4
plot(x, [f_1(x), 4 * x - 4], 'x', 'f(x)', legend=['f(x)', 'Tangent line (x=1)'], xlim=[0.1, 2], ylim=[-2,4])
练习 2
$$\nabla_{\mathbf{x}} f(\mathbf{x}) = \bigg[6x_1, 5e^{x_2}\bigg]^\top$$
练习 3
$$\nabla_{\mathbf{x}} f(\mathbf{x}) = \frac{1}{2}\bigg[\frac{2x_1}{\|\mathbf{x}\|_2},\cdots ,\frac{2x_n}{\|\mathbf{x}\|_2}\bigg]^\top = \frac{\mathbf{x}}{\|\mathbf{x}\|_2}$$
练习 4
$$\frac{\partial u}{\partial a} = \frac{\partial u}{\partial x} \frac{\partial x}{\partial a} + \frac{\partial u}{\partial y} \frac{\partial y}{\partial a} + \frac{\partial u}{\partial z} \frac{\partial z}{\partial a}, $$
$$\frac{\partial u}{\partial b} = \frac{\partial u}{\partial x} \frac{\partial x}{\partial b} + \frac{\partial u}{\partial y} \frac{\partial y}{\partial b} + \frac{\partial u}{\partial z} \frac{\partial z}{\partial b}. $$
