Apprendre The Chain Rule in Multiple Variables

Section 1. Chapitre 6

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When working with functions of several variables, you often encounter situations where variables themselves depend on other variables. The chain rule allows you to differentiate composite functions by systematically accounting for how changes in one variable affect others. Suppose you have a function $f(x, y)$ where both $x$ and $y$ are themselves functions of $t$ , such as $x = g(t)$ and $y = h(t)$ . The composite function is then $F(t) = f(g(t), h(t))$ . To find the derivative of $F$ with respect to $t$ , you apply the multivariate chain rule:

\frac{dF}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}

This formula means you take the partial derivative of $f$ with respect to each variable, then multiply by the derivative of that variable with respect to $t$ , and sum the results. This approach generalizes to functions with more variables and more layers of composition.

For example, if $f(x, y) = x^2 + y^2$ , $x = \sin(t)$ , and $y = e^t$ , you can compute the derivative of $F(t) = f(x(t), y(t))$ by first finding the partial derivatives of $f$ , then the derivatives of $x$ and $y$ with respect to $t$ , and combining them using the chain rule as shown above. This process is fundamental in multivariate calculus, especially when modeling systems where several quantities depend on each other.


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import sympy as sp

# 1. Define symbols
x, y, t = sp.symbols('x y t')

# 2. Define the outer function and inner functions
F = x**2 + y**2
x_t = sp.sin(t)
y_t = sp.exp(t)

# 3. Compute partial derivatives of F
dF_dx = sp.diff(F, x)
dF_dy = sp.diff(F, y)

# 4. Compute derivatives of inner functions with respect to t
dx_dt = sp.diff(x_t, t)
dy_dt = sp.diff(y_t, t)

# 5. Apply the multivariable chain rule explicitly
# dF/dt = (dF/dx * dx/dt) + (dF/dy * dy/dt)
# We must substitute x and y with their functions of t before multiplying
partial_x_term = dF_dx.subs(x, x_t) * dx_dt
partial_y_term = dF_dy.subs(y, y_t) * dy_dt

dF_dt = partial_x_term + partial_y_term

# 6. Evaluate at a specific point (e.g., t = 0)
t0 = 0
result = float(dF_dt.subs(t, t0))

print("Symbolic derivative dF/dt:", dF_dt)
print("The derivative dF/dt at t = 0 is:", result)

Tâche

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Apply the multivariable chain rule to compute the total derivative $\frac{dF}{dt}$ of the composite function $F(x, y) = x^3 + y^3$ , where $x(t) = \cos(t)$ and $y(t) = \ln(t + 2)$ .

Define the partial derivative $\frac{\partial F}{\partial x}$ utilizing the x_val variable.
Define the partial derivative $\frac{\partial F}{\partial y}$ utilizing the y_val variable.
Compute the time derivative $\frac{dx}{dt}$ utilizing the numpy sine function.
Compute the time derivative $\frac{dy}{dt}$ utilizing the t variable.
Assemble the chain rule components into the dF_dt variable utilizing the previously calculated partial and time derivatives.

Solution

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Section 1. Chapitre 6

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