Chain rule for multivariable functions

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Intros
Lessons
  1. Chain Rule for Multivariable Functions Overview:
  2. A Review of Chain Rule
    • What is the chain rule?
    • h(x)=f(g(x))h=f(g(x))g(x)h(x)=f(g(x))\to h' =f'(g(x))g'(x)
    • alternate notation: dtdt=dfdxdxdt\frac{dt}{dt} = \frac{df}{dx} \frac{dx}{dt}
  3. 1st Case of Chain Rule for 2 Variable Functions
    • z=f(x,y),x=g(t),y=h(t)z=f(x,y), x=g(t), y=h(t)
    • Then dzdt=dfdxdxdt+dfdydydt \frac{dz}{dt} = \frac{df}{dx} \frac{dx}{dt} + \frac{df}{dy} \frac{dy}{dt}
    • An example
  4. 2nd Case of Chain Rule for 2 Variable Functions
    • z=f(x,y),x=g(s,t),y=h(s,t)z=f(x,y), x=g(s, t), y=h(s, t)
    • dzdx=dfdxdxdx+dfdydyds\frac{dz}{dx} = \frac{df}{dx} \frac{dx}{dx} + \frac{df}{dy} \frac{dy}{ds}
    • dzdt=dfdxdxdt+dfdydydt \frac{dz}{dt} = \frac{df}{dx} \frac{dx}{dt} + \frac{df}{dy} \frac{dy}{dt}
    • An example
  5. Using a Tree Diagram for Chain Rule
    • Easier to find the chain rule for multivariable functions
    • What it looks like
    • Couple Examples
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Examples
Lessons
  1. Finding the Chain Rule for 2 Variable Functions
    Given that z=sin(xy),x=t3,y=1tz=\sin (xy), x= t^3, y=1-t, find dzdt \frac{dz}{dt} .
    1. Given that z=ln(x2+y2),x=s2+t2,y=2s+t z= \ln (x^2+y^2), x=\sqrt{s^2+t^2}, y=2s+t , find dzds\frac{dz}{ds} .
      1. Finding the Chain Rule for Implicit Functions
        Given that xy2+cos(x2y)=3xy^2+ \cos (x^2 y)=3, find dydx\frac{dy}{dx}.
        1. Given that exz+ln(xy)=x2y3e^{xz}+ \ln (xy)=x^2y^3, find dzdx\frac{dz}{dx}.
          1. Using the Tree Diagram to get the Chain Rule
            Suppose w=f(x,y,z),x=g(s,t),y=h(s,t)w=f(x, y, z), x=g(s, t), y=h(s, t), z=i(s,t)z=i(s, t). Use the tree diagram and use the chain rule to find dwdt\frac{dw}{dt}.
            1. Suppose w=f(x,y,z),x=g(r,s,t),y=h(r,s,t),z=h(r,s,t),r=a(q).w=f(x, y, z), x=g(r, s, t), y=h(r, s, t), z=h(r, s, t), r=a(q). Use the tree diagram and use the chain rule to find dwdq\frac{dw}{dq}.
              Topic Notes
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              Notes:


              Suppose we have a function y=f(g(x))y=f(g(x)). Then the chain rule is:

              y =f(g(x))g(x)y\ = f'(g(x))g'(x)

              We can rewrite this using an alternate notation:

              dydx=dfdgdgdx \frac{dy}{dx} = \frac{df}{dg} \frac{dg}{dx}

              Now if we were to change gxg \to x and xtx\to t, then we have the chain rule to be:

              dydt=dfdxdxdt\frac{dy}{dt} = \frac{df}{dx} \frac{dx}{dt}

              Why do we want this alternate notation? Because it relates to the chain rule for 2 variable functions.

              1st Case of Chain Rule for 2 Variable Functions
              Suppose we have z=f(x,y)z=f(x,y), x=g(t)x=g(t), and y=h(t)y=h(t), then the chain rule (derivative of zz in respect to tt) is:

              dzdt=dfdxdxdt+dfdydydt\frac{dz}{dt} = \frac{df}{dx} \frac{dx}{dt} + \frac{df}{dy} \frac{dy}{dt}


              2nd Case of Chain Rule for 2 Variable Functions
              Suppose we have z=f(x,y)z=f(x,y),x=g(s,t) x=g(s, t),y=h(s,t) y=h(s, t), then there are 2 chain rules.
              The derivative of zz in respect to ss is:

              dzdx=dfdxdxdx+dfdydyds\frac{dz}{dx} = \frac{df}{dx} \frac{dx}{dx} + \frac{df}{dy} \frac{dy}{ds}

              The derivative of zz in respect to tt is:

              dzdt=dfdxdxdt+dfdydydt \frac{dz}{dt} = \frac{df}{dx} \frac{dx}{dt} + \frac{df}{dy} \frac{dy}{dt}


              Using a Tree Diagram for Chain Rule
              Tree diagrams are very useful when finding the chain rule for multivariable functions with more than 2 variables.
              For example, suppose we have w=f(x,y,z),x=g(s,t,r),y=h(s,t,r)w=f(x,y,z), x=g(s,t,r), y=h(s,t,r) and z=i(s,t,r)z=i(s,t,r), and we want to find dwdt\frac{dw}{dt}.
              We can write the tree diagram below like this:
              chain rule tree diagram
              Then we will multiply all the connected derivatives, and sum them up to have:

              dwdt=dfdxdxdt+dfdydydt+dfdzdzdt\frac{dw}{dt} = \frac{df}{dx} \frac{dx}{dt} + \frac{df}{dy} \frac{dy}{dt} + \frac{df}{dz} \frac{dz}{dt}