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Chain Rule for Functions of Several Variables Calculator

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Example
Created on 2024-06-20Asked by Theodore Williams (Solvelet student)
Use the chain rule to find dzdt \frac{dz}{dt} where z=f(x,y) z = f(x,y) , x=t2 x = t^2 , and y=sin⁡(t) y = \sin(t) .

Solution

Step 1: Identify the functions z=f(x,y) z = f(x,y) , x=t2 x = t^2 , and y=sin⁡(t) y = \sin(t) . Step 2: Use the multivariable chain rule: dzdt=∂z∂xdxdt+∂z∂ydydt. \frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}. Step 3: Compute dxdt \frac{dx}{dt} and dydt \frac{dy}{dt} : dxdt=2t,dydt=cos⁡(t). \frac{dx}{dt} = 2t, \quad \frac{dy}{dt} = \cos(t). Step 4: Substitute these into the chain rule formula: dzdt=∂f∂x⋅2t+∂f∂y⋅cos⁡(t). \frac{dz}{dt} = \frac{\partial f}{\partial x} \cdot 2t + \frac{\partial f}{\partial y} \cdot \cos(t). Step 5: Conclusion. The derivative is: dzdt=2t∂f∂x+cos⁡(t)∂f∂y. \frac{dz}{dt} = 2t \frac{\partial f}{\partial x} + \cos(t) \frac{\partial f}{\partial y}. Solved on Solvelet with Basic AI Model
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1. Find the gradient of the func f(x,y,z)=sin⁡(xy2) f(x, y, z) = \sin(xy^2) at the point (1,−1,0) (1, -1, 0) 2. Calculate the partial derivative ∂z∂t \frac{\partial z}{\partial t} of the function z=f(x(t),y(t),t) z = f(x(t), y(t), t) given x(t)=et x(t) = e^t , y(t)=t2 y(t) = t^2 , and t=1 t = 1 .,
DefinitionThe multi-variable chain rule is a generalization of the chain rule in single-variable calculus to functions of several variables. It says that if there is a function f of several variables u, v, and if each variable in u, v depends on another set of variables x,y, then you can compute the partial derivatives of f with respect to x,y in terms of the partial derivatives of u,v wrt x,y. The chain rule of multivariate functions is a crucial tool and frequently used when finding the derivative of composite functions in multivariable calculus, in physics, engineering, and optimization problems with many parameters. Example: In economics, it is common to use the production function f(x,y) to describe the output of a firm as a function of the labor input x and capital input y, and the chain rule to compute the marginal product of labor and capital (which gives the impact of changing labor and capital inputs on output slightly )
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