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Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Partial Derivatives
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus Partial Derivatives
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
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Calculus
Concept Version 8
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The Chain Rule

For a function UUU with two variables xxx and yyy, the chain rule is given as dUdt=∂U∂x⋅dxdt+∂U∂y⋅dydt\frac{d U}{dt} = \frac{\partial U}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial U}{\partial y} \cdot \frac{dy}{dt}​dt​​dU​​=​∂x​​∂U​​⋅​dt​​dx​​+​∂y​​∂U​​⋅​dt​​dy​​.

Learning Objective

  • Express a chain rule for a function with two variables


Key Points

    • The chain rule can be easily generalized to functions with more than two variables.
    • For a single variable functions, the chain rule is a formula for computing the derivative of the composition of two or more functions. For example, the chain rule for $f \circ g (x) ≡ f [g (x)]$ is dfdx=dfdg⋅dgdx\frac {df}{dx} = \frac {df}{dg} \cdot \frac {dg}{dx}​dx​​df​​=​dg​​df​​⋅​dx​​dg​​.
    • The chain rule can be used when we want to calculate the rate of change of the function U(x,y)U(x,y)U(x,y) as a function of time ttt, where x=x(t)x=x(t)x=x(t) and y=y(t)y=y(t)y=y(t).

Term

  • potential energy

    the energy possessed by an object because of its position (in a gravitational or electric field), or its condition (as a stretched or compressed spring, as a chemical reactant, or by having rest mass)


Full Text

The chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if fff is a function and ggg is a function, then the chain rule expresses the derivative of the composite function $f \circ g (x) ≡ f [g (x)]$ in terms of the derivatives of fff and ggg. For example, the chain rule for f∘gf \circ gf∘g is dfdx=dfdgdgdx\frac {df}{dx} = \frac {df}{dg} \, \frac {dg}{dx}​dx​​df​​=​dg​​df​​​dx​​dg​​.

The chain rule above is for single variable functions f(x)f(x)f(x) and g(x)g(x)g(x). However, the chain rule can be generalized to functions with multiple variables. For example, consider a function UUU with two variables xxx and yyy: U=U(x,y)U=U(x,y)U=U(x,y). UUU could be electric potential energy at a location (x,y)(x,y)(x,y). The motion of a test charge on the xyxyxy-plane can be described by x=x(t)x=x(t)x=x(t), y=y(t)y=y(t)y=y(t) where ttt is a parameter representing time ttt. What we want to calculate is the rate of change of the potential energy UUU as a function of time ttt. Assuming x=x(t)x=x(t)x=x(t), y=y(t)y=y(t)y=y(t), and U=U(x,y)U=U(x,y)U=U(x,y) are all differentiable at ttt and (x,y)(x,y)(x,y), the chain rule is given as:

dUdt=∂U∂x⋅dxdt+∂U∂y⋅dydt\displaystyle{\frac{d U}{dt} = \frac{\partial U}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial U}{\partial y} \cdot \frac{dy}{dt}}​dt​​dU​​=​∂x​​∂U​​⋅​dt​​dx​​+​∂y​​∂U​​⋅​dt​​dy​​

This relation can be easily generalized for functions with more than two variables.

Scalar Field

The chain rule can be used to take derivatives of multivariable functions with respect to a parameter.

Example

For z=(x2+xy+y2)1/2z = (x^2 + xy + y^2)^{1/2}z=(x​2​​+xy+y​2​​)​1/2​​ where x=x(t)x=x(t)x=x(t) and y=y(t)y=y(t)y=y(t), express dzdt\frac{dz}{dt}​dt​​dz​​ in terms of dxdt\frac{dx}{dt}​dt​​dx​​and dydt\frac{dy}{dt}​dt​​dy​​:

dzdt=ddt(x2+xy+y2)1/2\displaystyle{\frac{dz}{dt} = \frac{d}{dt}(x^2 +xy+ y^2)^{1/2}}​dt​​dz​​=​dt​​d​​(x​2​​+xy+y​2​​)​1/2​​

=12(x2+xy+y2)−1/2ddt(x2+xy+y2)\displaystyle{\,\,\,\quad= \frac{1}{2}(x^2 +xy + y^2)^{-1/2}\frac{d}{dt}(x^2 +xy+ y^2)}=​2​​1​​(x​2​​+xy+y​2​​)​−1/2​​​dt​​d​​(x​2​​+xy+y​2​​)

=12(x2+xy+y2)−1/2(ddt(x2)+ddt(xy)+ddt(y2))\displaystyle{\,\,\,\quad=\frac{1}{2}(x^2 +xy+ y^2)^{-1/2}\left(\frac{d}{dt}(x^2) + \frac{d}{dt}(xy) +\frac{d}{dt}(y^2) \right)}=​2​​1​​(x​2​​+xy+y​2​​)​−1/2​​(​dt​​d​​(x​2​​)+​dt​​d​​(xy)+​dt​​d​​(y​2​​))

=(x+12y)dxdt+(y+12x)dydtx2+xy+y2\displaystyle{\,\,\,\quad= \frac{ \left(x+\displaystyle{\frac{1}{2}} y \right)\displaystyle{\frac{dx}{dt}} + \left(y+\frac{1}{2} x \right) \displaystyle{\frac{dy}{dt}}}{\sqrt{x^2 +xy+ y^2}}}=​√​x​2​​+xy+y​2​​​​​​​(x+​2​​1​​y)​dt​​dx​​+(y+​2​​1​​x)​dt​​dy​​​​

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